States that try to pass laws that one must use bathroom of the gender listed on one's birth certificate create a problem. Some transgender people are post operation and look more like the gender they are becoming than the gender listed on their birth certificate. Seems like the easiest solution, short of single occupancy facilities, is to use the bathroom of the gender you most look like currently. Determining what gender someone most looks like is best done by the people in the situation. Not easy to define at the government level.
Here's some more thoughts on the subject.
The binary system of men's and women's facilities was developed before we realized how fluid gender can be. Also before we took gay people into account. It was supposedly designed, I guess, to maintain modesty in separation of opposite sexes. New knowledge about our sexual natures is tossing a bit of a monkey wrench into that system. As that system becomes more stressed, or compromised, some folks fear that the illusive predator hiding under the bed will be able to take advantage of the situation. Not necessarily a fear of the transgender folks, themselves, but fear of what could happen as a system that so many folks got used to is being buffeted by the winds of change. Makes me think about a film I saw, when I was in high school, called Future Shock, by Alvin Toffler. As new information and change accelerate in society, a lot of people long for going back to what, I think, was described in the film as "Bonanza Land;" like the old TV show Bonanza.