Firstly the act of believing you are another sex is a mental illness.
Something like that for a kid is ridiculous and if you are a kid and you believe your another gender you are seriously ill. There is no scientific basis for one person being another gender. My advice is if you want to use the bathroom go in the disabled toilet or the one of your birth, tax payer should not be penalized for your stupid decisions.
Do you consider privately owned places as "public facilities"? If so then its up to the owner what their policy is, if they want one bathroom or none its their choice in my view.