This is simply a place where the law does not need to be. There are already laws that protect girls and women from perverted men.
Nothing about sharing a bathroom with people of any gender is a threat to people of any other gender. In fact it would be more democratic because the line outside the women's bathroom is usually longer. I know plenty of times that impatient women who would use the men's room and vice versa. Did I immediately phone the police to say that I came out of a stall to see a woman washing her hands in the men's room and that I had been bathroom-violated? Of course not, because I'm not a moron.
Bathroom door signs are not a legal issue. They are put there by the owners of those bathrooms, and if bathroom gender enforcement should be done, it should be done by the owners of those bathrooms. If I put a "women-only" sign on all the bathrooms in my home and then enter those bathrooms, have I commit a crime?
I think it reasonable that USING ANY BATHROOM for intended biological purposes by anyone of ANY GENDER without misconduct is not a criminal act. Unwanted sexual contact in the bathroom (or outside of the bathroom) is a criminal act and especially heinous if it involves children. However that has nothing to do with trans people. In fact, no trans person has ever been arrested for unwanted sexual activity in a bathroom. However there have been many senators involved in illegal bathroom sexual shenanigans. Perhaps it would be more fitting then to create a law to ban Republican lawmakers from public restrooms: