Many schools don't have gender neutral toilets, though some are gradually adding them, either as disabled toilets, or a single cubicle bathroom around somewhere. Though even with disabled toilets, unless you have a visible disability, it can be hard to use it.
I understand the issue you have. At school, I'd get picked on in the girls, and I'm sure it'd have been worse in the boys, so I didnt go to the toilet at school. It was uncomfortable, and it's kinda unhealthy. I know it sucks too.
I think your best bet is to talk to a teacher that you trust about it. If you are too scared/nervous to come out, you could perhaps just say you can't use the boys toilets for a personal reason, and see if they'll let you use another toilet, like the gender neutral one, even if it's not particularly for students. It's worth asking for the sake of your bladder.
Goodluck
