If you insist that trans women go to the men's room, then trans men have to go the the ladies, yes?

Such laws are all about enforcement realities. There aren't going to be police outside the door checking plumbing. Trans people who look like their trans sex won't be challenged by other users of the space, so the law isn't really relevant to them.

I distrust studies of the sort linked. Changing levels of awareness change the behavior of people. The linked studies revealed the effect of certain laws in certain places over a limited time period in a liberal state when trans issues were not au courant. It's like universal mail-in voting. It works once in a pandemic, but if it were the norm, corrupt pols would eventually find a way to intimidate voters. A study from 2020 would reveal little or no fraud, but it wouldn't prove anything about the dangers of the device.

There is no easy answer here. The point of my comment was not to support the law so much as to focus attention on its strongest argument.