I don't trust Labour on this. My local candidate has a 'Trans Ally' text and flag in her bio and refused to meet to discuss gender issues.
I voted Labour 1992-2017. The gender wars have just opened my eyes to the myriad ways the Labour party is institutionally misogynist (sex work is work, liberal crime policies, safeguarding fails, refusal to listen to women and outright sex discrimination e.g. Glasgow, Nottingham, closing down of free speech). Not just self-ID.
And bore off with 'Tories are just as bad / worse for women'. It's so puerile. I'm left wing, I will never vote Tory. I expect Labour to do better than them - but given they are not going to change tax policies, the two child cap, public investment or indeed do anything that will benefit women and families, I'd personally rather have the Tories who do at least allow public discussion and debate among their cabinet on this issue.