In terms of why the whole ‘feminist establishment’ supports transgenderism, one reason may quite simply be because of what happens to those who do not.
Believers in genderist ideology are ferocious in going after those who show even a hint of disagreement, so this has likely contributed to a false consensus on trans issues within feminist orgs and other institutions.
But this still leaves us with the question of why so many of the people fomenting this climate of fear are women themselves. I think it does go back to the idea raised by Biggs that many feminists seem prepared to defend extreme social constructionism to the death - to the point where they are happy to hand-wave or even facilitate truly awful consequences for women and children.
Pure social constructionism does lead to an assumption that humans, unlike other animals, are outside nature, and in that sense feminism has contributed to the situation summed up by Arnold:
there are people who right now seek to deny children the opportunity to ever experience the intricate, finely tuned wonder that is the human body of either sex. That there are people so arrogant they presume they can replicate or improve upon the rightful inheritance millions of years of evolution have bequeathed.
It is astonishing that so many people glibly accept the idea that there is such a thing as 'the wrong puberty' for a child and that doctors can re-engineer sexual biology so the child actually undergoes the puberty of the other sex. But if you accept that girls and boys are basically interchangeable, perhaps you are also easily seduced by the idea that their pubertal development is too. It takes a delusional level of biology denial, and, as Arnold says, an arrogant disregard for nature, to swallow this. But many educated people do. (Usually the same cohort that scorns climate change deniers!)
Of course there is a glaring contradiction at the heart of this, in that the notion of 'trans kids' is ALSO built on the assumption that boys and girls have different 'essences' and it's possible for a girl or boy essence to be 'trapped in the wrong body' and thus require medical correction.
The cognitive dissonance of the two simultaneously held positions - boys and girls are not biological categories but psychic essences! But also there is no meaningful difference between boys and girls and thus it's both easy and liberating to let them 'choose' to be the other sex if they want! - is characteristic of genderist ideology and a big part of why it has hoodwinked so many people.
Some feminists think that if you concede that men and women have evolved some psychological differences (e.g, propensity towards aggression, sexual behaviour), you are endorsing the 'feminine/masculine essence' theory and perhaps also implicitly endorsing transgenderism. On the contrary, if you have respect for human sexual evolution as a deep-time phenomenon that is reflected in every cell of the body, then you understand that being a boy or a girl or a man or a woman is a matter of embodied existence - there is no 'essence' that can be 'wrongly' housed in the other body.
Nor does this position preclude acknowledging the shared humanity of men and women, or individual variance within the sexes (saying that men have a greater tendency towards violence isn't the same as asserting that all men are inevitably violent), or examining the role of culture in shaping behaviour.