I think wanting good evidence is a very sound approach. (And also remembering that absence of evidence is not evidence of absence!)
But at the same time, this is at heart a philosophical argument, between those who (like me) see being a woman as a biological category, and those who see it as a "performative" or socially constructed category.
I see sex as a contingent fact about one's biology (which has no impact on one's personhood, but does have impacts on one's lived experience in terms of a whole load of things from how to play a rugby game fairly and relatively safely through to one's risk of being sexually assaulted, and the asymmetric consequences of that - e.g. both men and women can be raped, but only women can get pregnant as a result of rape).
Someone who believes in inner gendered personalities or souls (thank you Layla Moran for being honest, albeit philosophically incoherent) thinks that womanhood lies in somehow possessing some intangible feminine essence or inner sensibility.
Take prisons as an example.
If you believe women are simply adult human females then it is a clear violation of the human rights of female prisoners to single sex spaces for safety and dignity to incarcerate them with male prisoners. The numbers of male prisoners -0.1%, 1%, 5%, 20% - aren't actually relevant. One is too many.
If you believe TWAW, then you think it's a violation of the human rights of transwomen not to incarcerate them with female prisoners (and bugger the human rights of those female prisoners...)
We as a society have to choose. This is one of those occasions where that fucking stupid "rights are not like cake" meme shows itself up for the completely brain-dead fuckwittery it is. Only one side can "win" this one. There is no half-way house, no way of having the cake and eating it.