It is kind of misleading to promise a “mens” room (etc.) which ain’t.
Personally, I think this is all about someone looking for an issue, trying to make some sort of stand on ‘their’ rights.
The main issue is using the facility. So, use the bathroom that creates less drama for the world. Some people have such victim issues that they want the world to recognize and bow to it. Can you imagine if we catered to every little situation that someone perceives needs tailoring to?
Keep it simple, use the family bathroom, and let’s get on with our day.