"Our choices already appear to be determined by physical events inside our brains. That is, we are essentially flesh-and-blood robots."

By 'appear' you mean by the lights of secular humanistic methodological naturalistic reductionist 'scientific' program.  Isn't it self-evident that if this is your method of investigation that this will be the 'apparent' conclusion?

What about your own experience of reality?  Does that count for nothing?  If you 'appear' to actually experience 'free will' doesn't that give the lie to what materialistic scientists tell you? 

The idea that our choices are determined is an epistemological poison pill.   On the basis you suggest, I am as 'determined' to reject your atheistic conclusions as you are 'determined' to accept them.  Truth is no longer relevant.

Mainly, though, I just don't understand why God's presence would somehow affect our ability to choose to disobey him any more than a child is robbed of free will by the presence of his or her parent. As we all know, kids can choose to disobey even when the parent is glaring at them and muttering angry noises.

Out of curiosity, do you have kids?

You're wrong here.  Or, you're missing the point.  Yes, kids can choose to disobey even in the presence of their parents, but that only makes their insolence all the more pronounced.  Moreover, if they do obey while you are standing their breathing down their necks there is no virtue to their obedience.

Consider this real life example from this morning.  Every morning I have to tell my son to look both ways and watch for cars.  Every morning he obeys- great, he's not insolent, so that's a plus- but it is not a pure free choice because he knows I'm standing over his shoulder.   (in fact, in this situation the goal is to create something that is not a choice at all, but rather a habit.  When he is 15 and sees his first brutal car accident the choice to look both ways will be truly free as he sees the value in the effort).

The presence of the authority figure doesn't necessarily eliminate all free volition.  It does, however, influence the value and quality of that choice.  I want my son to clean his room on his own without me telling him to.  If I have to stand over him with threats, he may clean his room, but it was coerced.  If he doesn't clean his room still, we've got a different problem.

This is all illustrated very well with the case of Pharaoh.  Before God made himself crystal clear, Pharaoh was less culpable.   But even as God slowly revealed himself, it says that Pharaoh hardened his own heart.  Finally, after the death of Pharaoh's own first born, he relented and let the Israelites go.  But was there any value to this 'free' choice?  Nope.  As illustrated by the fact that less then a day after he changed his mind and gave chase to the Israelites.

If God had never revealed himself at all, Pharoah would have always had a measure of excuse.  The more God revealed himself, the more Pharaoh had to make a choice.  Finally, he hardened his own heart, despite clear evidence of God's operation, to such an extent that if God hardened his heart there was no fundamental difference.

Every revelation of God creates the same sort of situation.  The same principle is visible in the New Testament, too, with how the Jews reacted to Jesus.

Atheists wonder why God doesn't reveal himself to them to 'clear things up once and for all' but like you said, just because the Father is present doesn't mean the Children obey.  It might be for your own good that God does not reveal himself that clearly.  If he did appear, maybe you'd write him off as a murderer or something and thumb your nose at him.  That kind of insolence will bring you no closer to salvation, I assure you.