Trans women are Trans women.
There is a difference biologically between males and females, yes?
Women (and I use that term to mean adult human female, not a set of behaviours or thoughts or feelings) have long been oppressed because of this difference. This is why you can't just say Trans women are women. Their experiences of life and their bodies are so very different to women's that lumping them together as the same group with the same issues and needs does both Trans and biological women a disservice.
I don't think it's bigoted to accept there is a physical difference and this affects the needs of each group. I don't think it's bigoted to say that 'biological woman' should be a protected characteristic as they are an oppressed group - this is exactly why anti sexism laws exist, because of the treatment of women historically and currently. In much the same way as anti racism laws exist because of the historical treatment of non white people and the ongoing effects it has today.
I'm fine with non females living their lives as a woman in the 'social construct' sense, if it is how they feel their personalities are better suited and it's the self expression that makes them feel themselves and happy then I have no problem with that. I agree that gender roles should have nothing to do with genitals - having a penis doesn't mean you should suppress a 'more feminine personality' if that's who you are, but the difference is I don't think someone should have to literally become a woman in every sense, thereby erasing the needs of biological women. Why can't males wear dresses and make up and be 'feminine' without it changing that they are males? (lots of women don't do those things, are they not women?) Why should those behaviours be female identifying and not just open for any person?
If someone really wants a surgical sex change and it makes them feel good about themselves I see no issue with that either, the same as if it was a boob job or nose job or getting loads of tattoos. Nothing wrong with body modification, as long as the person is a consenting adult that is fully informed.
Regardless, the difference between Trans and biological women still needs to be protected in the areas that it matters - women's crisis centres and women's sports, for example. If that means a third space or category needs to be created then so be it. Personally, I think we should be working instead to smash the social constructs that defines women and men in these limiting ways rather than propping them up, which I feel the Trans movement is doing in lots of ways.