No.
First for legal reasons. Even if I were to accept that womanhood was a social construct (which I don't, but let's entertain the thought for the purposes of the debate), self ID would still be a bad idea. Think of all the other social constructs which carry practical, legal, real-world implications. Money, marriage, nationality to name but three. All of them are subject to gate keeping. You cannot just scribble "I promise to pay the bearer on demand of the sum of 10 pounds" on an old bus ticket and expect to pay for things with it in a shop. You cannot simply self-ID as married to get a married person's tax allowance. You cannot say "I identify as British even though I was born in Russia and have no British ancestors whatsoever" and expect the home office to say "Oh, right you are then..."
All these require objective standards against which claims of their truth or falsity can be judged. Self ID, in contrast, hinges on us believing the assertion of one individual person, with no evidence whatsoever admissible to the contrary, on the naive belief that no-one would ever lie. That is no basis for law.
Second, my objections are practical. Like many others, I started as a live-and-let-live sort of person. I'm the sort of person who campaigned against clause 28 and wrote letters to the PM defending right of residency for EU citizens in Britain post Brexit. But slowly that has been eroded by the behaviour of trans extremists. Trans extremists lost some of my good will when they called me TERF and likened me to Nazis. They lost more when they invited me to "suck their dick" on International Women's Day. They lost the last vestiges of it when they punched a 60 year old woman engaging peacefully in the right to free association and political expression, at Speaker's Corner of all places. This complete loss was confirmed when they attacked a woman trade-unionist on a picket line. It was set in stone when they made bomb threats, bomb threats which led the police to the discovery of all too real bomb making materials.
These are exactly the sort of people who have demonstrated, by words and actions, that they are the very last sort of people who should be allowed into women's safe spaces in any circumstances whatsoever.
My answer is NO.
NO.
NO.
Have you got that yet?
NO.