I see campaigning for a third space as a way of keeping single sex spaces for women.
First you have to spend all of your time convincing our legislators why we need single-sex spaces. Which now involves trying to persuade them that international human rights law continues to apply as written. Or that UK equality law has been misinterpreted and then misrepresented from the beginning. Or that allowing lobbying organisations undue influence over regulators is extremely undemocratic. Or that sex exists. Or that enshrining in law the ludicrous notion that one's preference for the sex stereotypes and sex role stereotypes associated with one or the other sex is present from birth and takes precedence over one's sex is not only regressive bullshit but also most damaging to females in particular. Or that medical treatment for children to which informed consent even for parents is impossible because risks are unknown is unethical and probably criminal practice.
I could easily list another 30 things we've been doing that take enormous amounts of time. Like writing submissions to umpteen consultations, which requires first teaching yourself equality law, reading hundreds of pages of consultation papers, checking references, reading papers then putting a coherent submission together, then following up, giving evidence etc etc etc.
Leafletting, letter writing, holding and attending public meetings. Networking. Planning campaigns, assessing failure and success then planning the next. Publicising on social media, blogs, websites, the media. Asking and being rejected and asking again and again and again to meet with the same politicians and ministers and public servants who happily allow themselves to be lobbied by trans rights organisations. Writing letters to councils, to companies, to charities, impact assessments, research papers, articles...
In addition to, as FloralBunting pointed out above, our paid work, raising our children, looking after our parents and all of the others things women do.
Thankfully there are now many of us, doing as much as we can in whatever time we can spare and we are nowhere near finished.
You may, if you wish, skip ahead and campaign for third spaces, but we're going to keep laying the groundwork first.