I am heavily involved in local governance. My advice with this is to be brutal and clinical.
It is only when you put their arse on the line that they stop bullshitting, and that inevitably involves financial liability of some sort. At the moment, there are no real world consequences for holding a position that ideologically opposes single-sex spaces -- so you have to nail it to the door.
Ask your MP if she will personally accept liability (political, financial, and legal) for any criminal offence that occurred due to the eradication of single-sex spaces in favour of mixed sex spaces in the climate created by the self-id legislation she supports.
Ask her if she will step down if such an incident occurs. If not, why not?
Ask her if she would be willing to publicly defend mixed-sex spaces in the aftermath of such an incident, remembering that the single-sex spaces that would be affected also extend to hospital wards, psychiatric wards, prisons, certain healthcare clinics (particularly sexual health), homeless and migrant refuges, and sleeping quarters in residential children's homes and elderly care homes.
Ask her if she would be prepared to print her opposition to single-sex facilities, particularly hospital wards, on her re-election campaign literature.
Ask her if she accepts her support of self-id legislation could be perceived as an attack on the rights of religiously conservative women, particularly BAME women. And if not, why not? Does she think that such women deserve to be informed that this is her position before the next general election?
Again, ask her if she thinks that her opposition to single-sex spaces could be construed as a form of institutional racism against women and children of Islamic, Hindu and Jewish heritage.
Ask if she accepts your letter as evidence that she had been made aware of the potential safeguarding issues inherent in self-id legislation.
Don't be wordy. Fire these points like bullets. In fact, put them in bullet points.