Short Answer to OP
It's a bit hard for me to understand the original question(s), so the majority of this answer is directed at the clarification that came from discussion in the comments. The short answer that precedes that is, by my lights, relatively uninteresting.
My short answer to the question of "whether the probability of a Christian god can reasonably be said to be infinitesimal", is that I couldn't say without clarification. It's certainly not the case that the mere fact of an infinitude of competing possibilities renders the possibility only infinitesimally likely. I don't think the absolute probability judgment is very easy to grasp or make sense of, so I would want to make a judgment of relative likelihood holding everything except for the proposition that the Christian god exists fixed for the sake of comparison.
I'd need to know whether the conception of the Christian god is of an intervening god who might leave detectable traces or more of a deistic "watchmaker" that starts the initial processes and then stays out of things. Since I've neither witnessed any miracles nor heard testimony I take to be reliable of such things occurring, I don't have any special evidence for God's existence.
So I'd evaluate the proposition that God exists according to the same standards I'd use to evaluate other theoretical posits, based on considerations like explanatory value. Based on the extraordinary powers typically ascribed to the Christian god, it seems that there are simpler explanations with equal explanatory value for the various phenomena I take to require explanation. So, I'd likely refrain from such a posit based on my evidence. To the extent that these more conservative explanations require less of reality, it seems that they would be more likely to be true. Of course, all of this is highly contingent on what you take to require explanation and what counts as a good explanation. Toss in some miraculous phenomena or require highly unified explanation and positing god's existence is a lot more attractive.
The Larger Issues
Following up on the discussion in the comments, I'm taking the relevant question to be
How do we know anything about the possibility of a reality beyond
what we directly experience through our senses, and can we judge the
relative likelihood of these extra-sensory possibilities?
with the specific extra-sensory possibility at issue being the existence of a Judeo-Christian god.
I'll break this into a few different issues concerning. This got very long, so I added a brief TL;DR to the longer sections.
How Do We Know What Exists?
The question of what we should believe exists is usually taken as a matter of determining the ontological commitments of our best theory of reality. There are a variety of criteria that we might use to compare and choose between theories, but simplicity and explanatory power are two frequently cited ones:
Simplicity: usually this is taken to be a matter of the fewest unexplained posits. For instance, a theory that posits only physical entities is simpler than one that posits physical and mental entities. Then some formulation of Ockham's razor would be appealed to in order to favor the simplest theory, all other things being equal.
Explanatory Power/Fruitfulness: another criteria for theory choice is how fruitful the theory is, its explanatory power. If a theory can explain more than competitors then, other things being equal, that theory is to be preferred. What exactly a good explanation consists in is itself a big area of study, some resources here would be the SEP articles on Scientific Explanation and Grounding as a form of Metaphysical Explanation.
An important challenge to any set of criteria for theory choice are worries about underdetermination. The worries floated here are that no matter what criteria you settle on, there will always be multiple mutually incompatible theories that satisfy your criteria just as well as one another. Given your criteria, the worry goes, all of these theories would have equal claim to being the theory you should adopt, yet only one can be true.
How Can We Have Knowledge of Things "Beyond our Senses"?
For this first question, note that there is nothing special about god here. Science posits and has posited all manner of entities that we have no -- and, in some cases never could have -- direct experience of. The relevant literature here would be the Philosophy of Science discussion on the nature of unobservables. Central to this debate has been the competing views on the aims of science. On one hand, Scientific Realists claim that science aims to provide us with theories that give a literally true description of the world, including whatever lies beyond what we can observe. For example, electrons were posited to explain the presence of a trail in a cloud chamber, but were not themselves observed directly -- we only observed their effect (the trail) not the electrons themselves. Scientific realists say that we should believe that electrons exist -- even assuming we can't observe them directly -- because the assumption of their existence provides the best explanation of the observable phenomena (so, they take it to be an assumption with high explanatory value and take that to be an indicator of its truth).
By contrast, Constructive Empiricists hold that the only aim of science is to provide an empirically adequate account of the observable phenomena and that we should remain agnostic about what unobservables there are. They deny that the explanatory value of the assumption of the electron's existence, e.g., provides us with reason to believe that there are electrons.
Note that these are both views about theories in the natural sciences. Neither has much to say about things like mathematical entities, which provide a better comparison to entities like a Judeo-Christian god since the truths of mathematics -- like the theological truths, if any -- are typically taken to be necessarily true.
TL;DR We gain knowledge of things beyond our direct sensory experience through abduction/inference to the best explanation. If the assumption that a certain thing exists provides a good explanation of the relevant phenomena, and there are no better competing explanations, that provides us with (defeasible) reason to believe such things really exist, despite our lack of direct acquaintance with them.
What About Necessary Beings?
The Judeo-Christian god, like mathematical entities, are typically taken to exist necessarily if they exist at all. What that means is that, assuming they exist, it is metaphysically impossible that they would fail to exist. Their non-existence might be epistemically possible, in that it is compatible with what we know, but these epistemic possibilities would not be genuine metaphysical possibilities. Rather, they would be impossibilities that merely seem possible due to our ignorance.
So, why posit such necessary existents? In the case of math, the idea typically goes that the truths of mathematics don't depend in any way on contingent features of reality and so, e.g., 2+3=5 would still be true even if absolutely nothing physical existed.
If something along those lines is right, then that gets us that mathematical truths are necessary truths, but why think there are numbers? Here the thought is that numerals are singular terms that, like names of people, refer to things. So, for "2+3=5" to be true, there must be some entities referred to by "2", "3", and "5" as well as some relation ("being the sum of") holding between these three numbers. If the truth is necessary, then the existence of these entities must also be necessary.
A similar line would go for god. If there are necessary truths that refer to god then god must exist of necessity. A common reason to think that truths about god are necessary is similar to the mathematical truths. If god exists outside of space and time, and existed prior to anything existing, then the existence of god doesn't seem to depend on any contingent features of reality. So, the story goes, god must exist of necessity.
In both cases, however, we would need some reason to think the relevant body of claims is true. In the case of math, we typically think that the utility of mathematics in describing and making predictions about the world gives us reason to believe the claims are true. Our experience of counting seems to give us reason to accept arithmetic, and measurement seems to commit us to something like a theory of real numbers. The truth of the corresponding mathematical theories seems to be the best explanation for the regularities we find in counting and measurement. So, we think these abstract theories are true.
In the case of god, you have the various arguments for the existence of god. You have said that you're assuming none of them are valid. In order for this to not be question-begging I'm going to assume that you mean that none of these arguments provide a deductively valid proof of the existence of god from obviously true premises (so none of these arguments are deductively sound). Then, we are left with the question of whether the existence of god is an assumption with explanatory value and whether it is the best explanation for the relevant phenomena. How you evaluate this will depend on what you take the class of phenomena to be explained to include. If you believe in miracles, which we'll assume by definition preclude a naturalistic explanation, then positing a supernatural being would seem to have high explanatory value. Even without miracles, you might think that assuming the existence of god provides the best explanation of the relevant phenomena, perhaps because it provides a highly unified theory of the phenomena.
In both cases, however, the entities being posited must be coherent which I take to involve something like Jo Wheeler's (1) and (2). While Set Theory is a highly fruitful mathematical theory, the original, naïve formulation was famously inconsistent. Modern set theory has largely settled on ZFC (Zermelo-Fraenkel set theory with the axiom of choice) as the appropriate formalization of the concept of "set", there are alternatives (e.g., Morse-Kelley set theory) as well as various extensions (e.g., the addition of large cardinal axioms). So, even if you think ZFC is true and so sets exist, that still leaves open exactly what the set-theoretic universe is like and what properties these sets must have. All of this is to say that what it takes to satisfy the criterion of being well-defined is not perfectly clear, even in the comparatively uncontroversial case of math. Similarly, while we believe ZFC is free of contradiction the results of Gödel show us that we cannot hope for any absolute proof of its consistency (and indeed, this holds for theories much weaker than ZFC).
Going to theories of god, the same constraints must be satisfied. The theory of god being posited must be consistent (even if we cannot prove it is so), and the concept of god on offer must be reasonably well-defined.
TL;DR The issues here are the same as with knowledge of things beyond our senses more generally -- we reason to their existence by means of inference to the best explanation. The additional ingredient here is that we also need reason to think that the truths we have inferred via inference to the best explanation are necessarily true, which requires separate argument and usual appeals to some sort of independence from the contingent features of reality that contingent entities beyond our sensory experience would not enjoy (e.g., if there were no physical things then there would be no electrons).
How Do We Know What's Possible?
Another issue concerns matters of the epistemology of modality, our knowledge of possibility and necessity. I'll be brief here since this answer is already obscenely long, but note that we take the inconsistency of a theory to show that the entities defined by such a theory couldn't possibly exist (at least as the theory describes them). Beyond that, a lot of our reasoning about what's possible falls into one of two categories:
Combinatorial Reasoning: We often think that if it's possible for two things to be true independently, and the two things don't contradict, then it's possible for them to be true jointly. If I could own a red shirt, and I could own blue shoes, then I could own both a red shirt and blue shoes. Spelling this out in more detail is a difficult matter but hopefully the intuitive idea is clear enough.
Analogical Reasoning: We often reason that if some scenario is possible, then relevantly similar are also possible (barring any obvious block to this). For instance, if it's possible for my hair to be brown then given that there are other people very much like me with blonde hair, it's possible for my hair to be blonde. How might this be blocked? Well, perhaps given the genetic makeup of my parents it would be impossible for them to have a blonde-haired child (I'm just using this as an example, I don't recall the genetics of hair color inheritance). If it's essential to me that I have the parents I actually have, then it's impossible for me to have had blonde hair. That putative essential trait would "block" this analogical inference.
How Do We Compare the Relative Likelihood of Different Possibilities?
Finally we get to the crux of the issue which is that, given a range of different possibilities, how do we determine which of them are most likely? I take this to be a question of determining and comparing the probability of a number of different possibilities. Here, the details regarding how probability is understood will matter.
There are three main options (quoting from the SEP article):
A quasi-logical concept, which is meant to measure objective evidential support relations. For example, “in light of the relevant seismological and geological data, it is probable that California will experience a major earthquake this decade”.
The concept of an agent's degree of confidence, a graded belief. For example, “I am not sure that it will rain in Canberra this week, but it probably will.”
An objective concept that applies to various systems in the world, independently of what anyone thinks. For example, “a particular radium atom will probably decay within 10,000 years”.
If probability is understood according to (3), then there is an objective fact of the matter about how probable different possibilities are. Since necessary truths have a probability of 1 (i.e., they are certain) and impossibilities have a probability of 0 (i.e., they are certainly false), it would not make much sense to ask about the probability of god's existence, if god is claimed to be a necessary existent. For each necessary existent the probability of its existence is either 1 or 0 and so propositions of the form "necessary existent X exists" can only be compared for (actual) truth and falsity.
But (3) isn't the interesting option here. How about (2)? This gets us into the realm of Bayesian Epistemolgy. This is concerned with the relation between your prior beliefs and a new hypothesis you're entertaining, particularly with the credence (i.e., degree of confidence) you should have in this hypothesis given (or conditional on) your priors (your beliefs and the confidence you have in them, with confidence assigned in a manner than obeys the laws of probability). Then what we're asking about is the relative likelihood of competing hypotheses given your priors. This can be done pairwise as shown by the Ratio Formula in section 4.2E in the Bayesian Epistemology SEP article:
[To] compare the effect of evidence E on two competing hypotheses, Hj
and Hk, without having also to consider its effect on other hypotheses
that may not be so easy to formulate or to compare with Hj and Hk.
From the first corollary above, the ratio of the final probabilities
of Hj and Hk would be given by:
Pf(Hj)/Pf(Hk) = [Pi(E/Hj) × Pi(Hj)]/[Pi(E/Hk) × Pi(Hk)]
If the odds of Hj relative to Hk are defined as ratio of their
probabilities, then from the Ratio Formula it follows that, in a case
in which change in degrees of belief results from conditionalizing on
E, the final odds (Pf(Hj)/Pf(Hk)) result from multiplying the initial
odds (Pi(Hj)/Pi(Hk)) by the likelihood ratio (Pi(E/Hj)/Pi(E/Hk)).
Thus, in pairwise comparisons of the odds of hypotheses, the
likelihood ratio is the crucial determinant of the effect of the
evidence on the odds.
So, it would make sense to compare any two of the infinitely many possibilities by this procedure as long as you fix on some particular set of evidence (i.e., set of beliefs and credences distributed among them). Of course, as a practical matter, carrying out this comparison is not possible since assuming a countable infinity of possibilities there would be an uncountable number of comparisons to run through. But it makes (mathematical) sense to compare the relative likelihood of these possibilities. Additionally, you might be able to "prune the search tree" and eliminate all but a small number of possibilities. Perhaps you and your interlocutor are agreed on everything except the one proposition that the Judeo-Christian god exists. Then you can compare the relative likelihood of that proposition and its negation. If your shared evidence lends support to either proposition, it will disconfirm the other.
Of course, even assuming you agree on what the priors are, you might disagree with respect to how your portion your credence among the beliefs that make up your priors. If that's the case then it might be that you cannot compare the likelihood that god exists, because it is highly likely given one set of priors and highly unlikely given another. At this point the only thing to do would be to argue that one of your has better priors than the other. Assuming both of you have probabilisticaly coherent priors, the only thing that could tip the scale here is if Objective Bayesianism (4.2F(b)) were true and there were some rational constraints beyond the probability axioms on how prior probabilities are constrained that gave one of you the "better" priors.
Finally, if (1) is the correct theory of probability, then your shared evidence would support each of the possibilities to some specific, objective degree. It's possible that many of the possibilities would be tied. It's also possible that given the complexity of the relations of evidential support, neither of you would known whether your shared evidence supports god's existence or god's nonexistence.
TL;DR The details will vary depending on how probability is understood, and on what norms are taken to govern the confidence we ought to have in a given possibility given our total evidence, but comparing the likelihood of these possibilities is often sensible, though often not practical. The sheer number of possibilities is not much of an obstacle, since we can usually reduce the issue to pairwise comparison of the relative likelihood of different possibilities. The biggest obstacle to meaningful comparison will not be the vast number of possibilities, but the vast difference in what you and your interlocutor take to be evidence and what you take that evidence to support.