Mental illness is not a special need, why should others carry the cost of their delusions? They should use the available facilities as per their real gender, not the one they made up.
Then again, assigned bathrooms aside, there are bigger issues than men using women bathrooms and women using men's bathrooms -
* Why are adults changing around kids anyway? you don't need to be transsexual to be a sex offender
* I understand that women don't want to see men's junk but there are plenty of men that don't want to see men's junk either - unisex bathrooms in public spaces are more practical and convenient. previously there were no baby-changing tables in men's bathrooms despite the amount of dad's that actually look after their kids in public areas
* If sexual assault/harassment is an issue, a label on a door isn't going to prevent that, a guy dressed as a woman committing an assault is going to get the same reaction/treatment as a man entering a female bathroom and committing an assault
* For kids protection, I understand the importance of separate bathrooms and dressing rooms but the issue comes down to how the adults handle that responsibility. A water polo coach was molesting boys in a high school, he never dressed up or told people he was a unicorn, he simply had access and exploited it. There could be a female water polo coach doing the same thing in the girls water polo team. Don't worry what people do with their junk and clothes, worry about who has access to your kid