I think you're asking about whether the abstract idea of a "model" is well-founded. For a given axiomatic theory of logic, a model is intended to be some set of objects that satisfies all the formal axioms.
In theory, a model allows us to assign a truth value to every logical proposition, whether or not that proposition can be proved from the axioms.
When you say "model-theoretic regress" I believe you are talking about the regress from talking about propositions and axioms as purely textual things, to talking about objects in the model, to which the textual propositions supposedly refer.
I agree with you that this is problematic, because of the Pragmatic Maxim.
Consider what effects, that might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.
The notion of a model is unjustified, because it is disconnected from any physical object that the mathematician works with. The objects a human mathematician directly interacts with includes the text of the proof, diagrams and images on paper or in their imagination, and other activity of their own neurons. The mathematician performs calculations involving these physical objects. The mathematician is probably limited to what a Turing machine can do, if we suppose that the mathematician's brain can be simulated on a Turing machine, which seems likely, or at least has not been ruled out.
The notion of a model - that yields a true or false answer to every question we ask in the logical theory, whether or not the question can be answered according to the formal axioms - goes beyond anything a Turing machine could do, and goes beyond anything the mathematician could do. The model "decides" everything. What practical connection does the mathematician have to the model? If the model says a certain proposition is true - though this proposition can be neither proved nor disproved by any axiom (or meta-axiom, or informal rule of thought) the mathematician knows - then in what practical way can this possibly matter to the mathematician?
To take a specific example, consider the continuum hypothesis. The continuum hypothesis is independent of ZFC; we may take the axioms of ZFC and assume additionally the continuum hypothesis, or we may take the axioms of ZFC and assume additionally the negation of the continuum hypothesis. Usually mathematicians do not declare whether they are taking the continuum hypothesis to be true or to be false. It does not matter to their work.
So, if they haven't even considered the continuum hypothesis, what model are they working in? No model of ZFC + (continuum hypothesis) is also a model of ZFC + (negation of continuum hypothesis). By declining to take a position on the continuum hypothesis, a mathematician cannot be working in any specific model, and thus their propositions cannot "refer" to any specific model.
Even if the mathematician does declare their stance on the continuum hypothesis for the purposes of a particular paper, there are an infinite set of other propositions, that, like the continuum hypothesis, are also independent of ZFC, and which the mathematician has not declared a stance on. So, again, the propositions they derive cannot refer to any specific model.
