Being Itself and quantifiers

I'm worried that formalizing a cosmological argument that stays true to the classical traditions of God being "beyond being and nonbeing" runs into an intrinsic difficulty. Any such argument must conclude (if it is to be a cosmological argument at all) with a statement like

∃xGx

where Gx means x is God, or whatever sort of proto-God we've proven at this point in the argument, like actus purus or a necessary existent. But if God is beyond being and nonbeing, that is, God is not to be numbered among what are merely beings and not Being, then it would seem that God isn't in the domain of our quantifier. Indeed, to say that ∃God would be to state that God exists as one more thing among all of the other things, which is obviously not how God is thought of.

So, if we wanted to respect God's transcendence in this regard, the more accurate way of speaking of God would be to say that

!∃xGx

but this conclusion is indistinguishable from a failure of the argument. Perhaps we could say that

∃x df. ∃1x ∨ ∃2x

and ∃1x corresponds to the creaturely mode of being and ∃2x corresponds to the divine mode of being. But then we'd run into problems with using ∃ in our premises instead of ∃1, which could prevent us from inferring to ∃2something.

Of course we could always say that God is beyond the grasp of any puny human system, but this is not a philosophically satisfying rebuttal.