What does 'behave as a woman' mean, David?
Whatever answer someone comes up with it will be different from how women behaved 100 years ago, and how they behave in other cultures. However far back you go, and however far away, women are female and men are male, however. Gender is not fixed, it is constructed. Sex is biological.
If a man wants to wear a dress, he can crack on, as far as I'm concerned, but wearing it doesn't give him a right to enter spaces reserved for women and girls. If we allow that, we have effectively erased the protection that feminists fought for, and have removed the right to privacy and dignity when naked, ill, restricted in movement or otherwise vulnerable for all women and girls. If anyone thinks that is ok, as mens' 'feelings' are more important than women's safety, I would ask them why they think these spaces were introduced in the first place?
We can all go to the pub, sit on a train, walk in the park wearing what we like and that's fine, but hospitals, changing rooms, prisons, care homes and DV hostels are different because they house vulnerable women and girls and should be off limits for men, however they 'present' themselves.