The skepticism at issue here falls under the heading of mysterianism or cosmicism. I doubt whether the possibility that anything is beyond our comprehension amounts to a proof that there's such a thing, but the analogies that establish the possibility are compelling in a dramatic sense (in Lovecraft or in Thomas Nagel's discussion of consciousness, for example).

We could certainly put our human spin on any great mystery, but how would this differ from the scenario in which you give a cat a computer and the cat walks all over the keyboard and meows at the screen?

The issue, I think, is whether we can answer every meaningful question in a way that passes muster, by meeting our standards for understanding and mastery. If we can comprehend something, we can potentially master it. So is there anything that exists that can't be mastered by any evolved intelligent life form? Such a thing would be divine or God.

But there are clearly two issues here, the semantic one of clarifying what's meant by "God," and showing that there's probably such an invulnerable being.