If there's a backlash that men no longer have spaces to themselves then men are going to have to do what women have done, and raise awareness and campaign for it. It's not the job of women to make sure men have everything they need before taking care of their own needs. Frankly at the moment, women are a bit busy.
Fwiw, I wholly sympathise.
Women want dedicated, sex based spaces used only by those of the female sex. If men want dedicated sex based spaces used only by those of the male sex, then I wholly understand and sympathise. There will then need to be additional spaces for those bothered about biological essentialism and stretching the bandwidth of sex who do not want to use their sex based space, who have changed their appearance to the point it would cause confusion in a sex based space, and who want all that freedom of expression. Great, those should be provided if wanted, and those people can campaign for them. There's a lot of very well paid charities involved and the civil service all with rainbow badges, they should have an easier time than men and women wanting single sex provision.
What they cannot do is deprive others of their sex based spaces for their own personal satisfaction of having removed them from others. While wittering about their personal extremist beliefs and how this somehow justifies removing need provision from others and requiring everyone to revolve around them. No thank you.