Transphobes are uncomfortable with someone of a different sex using the same restroom because they make that determination based on sex
Changing the label doesn't change the logic. Transgendered people who can't handle using a restroom with other people of the same physical sex as themselves are just as phobic. Yet somehow that's ok, but the other isn't? Why?
There is nothing preventing that kid from using the boy's bathroom except his/her own fears and hangups. None at all. Yet, we're expected to require that everyone else set aside their hangups about sharing a restroom with someone of the opposite sex on hold because a very tiny percentage of the population feels uncomfortable the other way around? That makes no sense at all.
If I declare that I'm made uncomfortable at the sight of women with clothing on, can I demand that we change the rules to require that they all walk around nude? I'm the minority here, right? So the fact that most women will be uncomfortable complying with what I want shouldn't matter. They're just bigots and haters of a minority. They simply don't understand the mental and emotional stress I go through every day having to deal with women around me who are wearing clothing! Argh... If the world doesn't fix this, I'll just have to commit suicide.
Silly? Yes. And so is the idea of transgenders imposing their own phobias and hangups on the rest of the society. They're the ones who have a disconnect between their physical sex and their psychological gender. The rest of us don't. They're going to have problems dealing with that regardless of what restrooms they're allowed to use. So we're looking at a ridiculous and largely unworkable demand that isn't really going to fix anything anyway. So how about we just not do it and tell the little boy/girl that the world works this way and that he/she will need to learn how to interact with that world as it is. It's not going to change for him/her.
There's no practical way to change the vast hosts of things in the world that will cause problems for a transgendered person. Better to set realistic expectations and move forward IMO. And guess what? The kids probably not going to get a pony either. That's just the way the real world is kid. Deal with it.