Elle, ignore arguments about who is a "woman" - that's just a word. Look at the facts.
Female people - the ones with female bodies - exist. This is undeniable.
Female people have been stucturally and culturally disempowered in favour of men in almost all cultures, including our own. In our culture we were unable to own property, unable to vote, denied bodily autonomy wrt rape and abuse, typically given less access to education, explicitly barred from many cultural and economic arenas and in the ones implicity barred in ways ranging from being made unwelcome and ignored, to being ridiculed to our faces or in the media, to being "allowed" but prevented in practice by lack of support for our physical needs or through lack of anyone else to take on childcare, up to verbal and physical assault.
A significant aspect to this legal and then social marginalisation was the cultural belief that since male and female bodies are different, so must be male and female minds, and therefore it was right and proper that female people were excluded from intellectual, political and economic spheres, because their minds were not right for them. Female minds were assumed to naturally better the domestic sphere and for supporting roles.
And although these things are no longer legally permitted and should be socially rejected, they still cast a long shadow on how we are perceived and the degree to which we are practically supported to participate in society on equal terms with men. This is also undeniable.
Those who believe in gender identities believe in this idea of a "female mind", but believe it can occur in either sex.
I'm not going to argue about whether "women" are the people with the female bodies or the people with the female minds because that is missing the point.
Clearly, since both groups exist (even if the latter has no fundamental physical existence and is simply a belief that has arisen among some people about themselves, it is no less real or valid or existing than religions which we respect and accomodate), the two concepts - having a female body, and having a "woman's mind" - both need their own language and their own rights and to be seen as separate things.
The groups overlap of course - plenty of female bodied people believe they share something meaningful mentally with trans women and I see no need to disbelieve them - but the word Woman cannot mean both groups at same time, nor can the same Woman-only spaces and protections cannot serve the needs of both groups simultaneously.
So my issue with the way Labour and TRA-aligned politicians in general are treating this is not about the giving of rights to trans women, it is about the taking away of the language and rights and indeed political, legal and social recognition of female people as a legitimate group.
Because regardless of whether trans women are "women" or not, we, the female-bodied people, still exist, and we experience social and economic consequences because of our sex that are nothing whatsoever to do with the "womanhood" experience of trans women.
We have a different history and cultural baggage from trans women, so we have different needs to trans women.
We, the female people, need the legal and political langaue and right to identiify oursleves as a separate group to male people, to talk about and collect data on our experiences and to agitiate for whatever rights, protections and opportunities we need to mitigate it and it is not an act of hate to say this.