That is not what "biological essentialism" means. But I'll put terminology aside and deal with your basic point.
There are observed physical facts that underly our concept of "sex", not the least of which being that it is an unmodifiable fact that one type of body gets pregnant and one type does not, and moving cultural classification around and even having no name for the difference at all will not change one iota which group an individual falls into.
However, as you say, how we think about those facts of "sex" and the significance and meaning we ascribe to them are socially constructed. We can observe or imagine societies that place less significance on sex, or draw lines on secondary characteristics in different ways, eg societies which subdivide males into Men and AnotherType of some sort. They don't turn male people into female in the biological sense but they do create different social meanings within that biological binary.
But all that notwithstanding, in this society we did construct the meaning of sex in this way, and that has material consequences for both sexes but for women, who bore the brunt of patriarchy and still live under its social structures and expectations, particularly.
So I reject utterly the idea that because the meaning and significance of sex is in part socially constructed, it is an irrelevance that can be ignored or undefined. It is as offsensive to me as a female person as it would be to tell someone with lived experience of racism that because race is culturally constructed their race is not relevant to their needs and experiences.
Women, in the original sex-based meaning, will continue to exist in a meaningful social and cultural sense for as long as they are identified as such in ways that have consequences.
And as long as that exists we have a moral right to say we are not the same as trans women, we are not two subsets of "women" with much in common and little to separate us, we are fully realised humans in our own right dealing with the consequences both nature and society have constructed around our bodies and we need our own name, our own spaces, our own political movement and our own voice to manage those consequences.
I get it. You think if society stops constructing meaning on sex differences wonen would be free of sexism. You think women only get treated badly because society keeps reinforcing that bodies matter. Hey, in the long term, and assuming appropriate mitigation of the asymmetric consequences reproduction, you are probably right. Women's spaces protect women in the short term but also maintain the idea of women's privacy as erotic and exciting for men. Catch 22.
But you remind me of the type of "progressive" who wants to free veiled women but is too scared of challenging men so brings in laws that veils can't be worn. He thinks he is freeing the women but all he is doing is controlling them a different way, forcing them to act out his idea of a better world from a place of vulnerability and expecting that somehow changes the behaviour of the people with the actual power. Why on earth would anyone think that would work?
Forcing women to pretend our body sex is not significant and has no consequences in a society where it clearly clearly does isn't Feminism, it's gaslighting and its just opening the door for men to further abuse, exploit and marginlise us.