A transgender woman is a woman, not a man. If you are a woman, you can use the women's room.
Have you known any trans people? They are almost always unnoticeable. They don't dress like women but look and act like men, they do everything to be what they feel like inside. Surgery, hormone therapy, therapy in general -- doing things to become their trans gender. You don't have people like Chris Helmsworth just throw on a dress one day and say "I feel like I'm a woman now."
And if a woman who looks like a man walks into a women's room, do you think people honestly aren't going to say anything? If Seth Rogen walked into a women's room, would you think nobody is going to say anything?
You asked about they have to PROVE they are trans? Why? You're making up situations to need proof when you don't. If you get the store manager and they ask the person to leave and it turns out they were trans, if they file a complaint nobody is going to jail; nobody is going to be fined. Honest mistakes are going to happen. Heck they already happen today with women dressing like men who are women, and vice versa. Are you going to demand to see the privates of every person who walks into a restroom that's you are unsure of?