Hope this doesn't get me deleted - I never post here and I'm not 100% on the rules.
I would join the chorus saying that you are biologically female. Therefore you use the ladies toilets. Because they are sex segregated not gender segregated.
This is my reasoning:
The reason for sex segregation is safety - gender identity of woman does not instantly remove the biologically male factors that create a risk for women. I believe that the majority of men are fine, and not a threat. The same, I'm sure, are the majority of trans women. However - the minority (of all males, not isolating trans women) that present a risk are real and present such a risk that sex segregated facilities are necessary. Males that are a risk and wish to invade women's spaces will use gender ideology to do so in a socially acceptable way. I am not saying that all men who nefariously enter women's spaces are trans - I am saying that predatory males will lie to get access!
As to identity. I am GC. My sex is female. I am biologically XX. My identity is 'grannyachingsshepherdshut' I do not have a gender identity because I believe I have a sex and that, as you say, gender identity is just perpetuation of harmful stereotypes.
Sometimes I wear makeup.
Sometimes I wear mens jumpers.
My male DP always takes the bins out.
I can drive a digger.
I accept men holding doors for me.
I hold doors for men.
Which of those makes me woman? Or man? In my opinion - they're all irrelevant.
I do not consider myself 'trans' (I am aware that stonewall definition probably would, but I also think they wouldn't want me even if I came with a free gold clock...)
I use the ladies loo.
I believe that harmful stereotypes are only destroyed by people breaking them. By people who don't wish to be constrained by stereotypes stepping outside of the polarity of male/female and identifying as agender, it feels that those stereotypes are perpetuated. A reasoning of 'I'm not girly, so that means I can't be a woman, I must be something else because all "women" are girly'.
I genuinely believe that the majority of people don't have a gender identity - they just know what sex they are.