I think there are different kinds of necessity at play in your argument that should be kept separate. (I may be repeating some of what you said, and some of what you and Conifold discussed in chat.)
If a proposition is true at all moments in time, let's call that, let's call that proposition "eternally true".
If a proposition is true in some possible world, that proposition is called "metaphysically possible". If a proposition is true in all possible worlds, that proposition is called "metaphysically necessary".
If a proposition is free from contradiction, that proposition is called "logically possible". And if it's a contradiction to deny it, that proposition is "logically necessary".
Logical necessity implies metaphysical necessity, and metaphysical necessity implies being eternally true.
The reverse implications do not generally hold. A proposition may be true for all moments in time, without being true in all possible worlds. An example might be "the physical universe was caused by a big bang", or "nothing is travelling faster than the speed of light", or some law of physics. But, even though such propositions are always true in our world, it doesn't mean that they must be true in all possible worlds. In some worlds, there might not have been a big bang, or faster-than-light travel may occur.
There could be metaphysically necessary truths, such as "God exists" or "everything has a sufficient explanation" which are not logically necessary, since denying that God exists or asserting that there are brute facts doesn't seem to lead to explicit contradiction. And likewise, there may be some logical possibilities that may not be metaphysical possibilities. For example, Saul Kripke has argued that identity statements with rigid designators such as "water is H2O" are metaphysically necessary. If he's correct, then "water is not H2O" is not metaphysically possible, despite being free from contradiction (ie. logically possible). I think debates about abstract entities, causation, and the nature of time are like this too. Philosophers will leverage intuitions and possible-worlds talk to argue for metaphysical truths, even though the negation of such truths may not be explicitly contradictory.
Anyways, I understand your argument as trying to infer God's metaphysical necessity from God's eternity:
- If the proposition "God exists" is eternally true, then the proposition is metaphysically necessary.
- If the proposition "God exists" is eternally false, then the proposition is metaphysically impossible.
- "God exists" is either eternally true or eternally false (since God cannot come into being or cease to exist).
- Therefore, "God exists" is either metaphysically necessary or impossible.
It seems to me that we can't in general infer a being's metaphysical necessity from that being's eternal existence, for reasons that I gave above. Is so, then additional justification needs to be provided for 1. and 2.
It occurs to me after rereading your question, that maybe you're just trying to infer God's eternal existence or eternal non-existence from it being impossible for God to begin to exist or cease to exist. If this is the case, I think the talk of modal logic and logical necessity are somewhat distracting.
It also seems that your notion of metaphysical necessity is not truth in all possible worlds, but about what actual-world entities are able to cause (ie. to you, "possibly P" means that "actual-world entities can, by their causal powers, bring it about that P"). That's fine, but just remember that God's being necessary or impossible means just that there is no causal chain that can bring it about that God doesn't exist, or that he does exist. God's necessity (in this sense) is no longer about conceivability or logical possibility. I mention this because it seems like the next natural step in your argument is to say something like "God isn't impossible" since God is free from contradiction, so God must be necessary. But this isn't what your notion of necessity means any more.
