"As far as I know, Aquinas didn't contend that we could understand Christ's human nature only analogically. And the Incarnation itself is still a mystery, so even if we understand univocally that Jesus did this or that, we still don't understand univocally how it is that God did this or that."

But if we predicate things about Christ's human nature, aren't we predicating something about God? We can predicate about God that he is such a being that his person can have a human nature and we mean that he can have it in a sense univical to how we have it (and it must be univical to how we have it, or else he has not assumed humanity in the way necessary for our redemption).

Consider (to adapt a fairly well-known example) a blind man who is told that "red is like the sound of a trumpet." He doesn't have a univocal understanding of red, and yet the analogical understanding seems to be effective enough for his purposes.

How could we or he know that it is "effective for his purposes"? We have no way of knowing whether he has gained any knowledge about the color in itself that actually corresponds to reality, nor does he. The analogy only works for the sighted--and then only weakly--because we have prior knowledge of redness and the sounds of horns. It seems the blind man might just as easily gain a misconception about the nature of redness when given such an analogy; and couldn't we as well gain grace misconceptions about God if we really are like a blind man being given analogies of color?

As far as I can see, our apparent ability to make such predications doesn't undermine Aquinas's primary point, which is that we don't have any direct knowledge of God's essence, what God is in and of Himself.

What if we rephrase the predication to "God, in and of Himself, has an essence such that he can take on a human nature (though the means of incarnation remain a mystery to us)"? For that matter, why not "God's essence, in and of itself, is such that finite creatures can't make non-analogous predications about it"? 

We could always ask him. But it doesn't matter; some analogies are closer than others, and I deliberately chose one that was rather a stretch just to make the point.

Asking him doesn't help at all, because the point is that no one knows, not even him, whether the analogy is helpful for conveying what redness really is. He can't compare his idea of redness to the actual thing, and we can't compare actual redness to his idea. So there's an unbridgable gap. And  this point holds even if we talk about a better analogy: If one were to call a sloppy person a "pig" the anology is very apt, but if I just don't know what a literal pig is then it doesn't matter how much of a stretch or not the analogy is, it simply can't convey useful information to me.