Remember these facilities are labelled by phenotype/identity not genotype.
That is a misrepresentation.
When "these facilities" aka women's provisions were established, the label Woman meant female bodied, to the level that was understood at the time which is roughly "born with visually female sex organs", and therefore subject to the social pressures, risks and indeed advantages that people who met that description faced. The facilities exist because of the people who needed them.
The concept that the word Woman might be repurposed as a label of a mixed sex identity was not on the radar at all.
It is utterly disingenous to portray these provisions as if they just exist in a vacuum for any group with the political and social power to claim the label Woman to take. The label is irreelevant. It is the needs of the people, not the label, that gave shape to the facilities that now exist, and those people were female. These are facilities based on female needs and males, regardless of their label, have no claim on them.
To look backwards from today, when Woman has (arguably, not universally, and in the face of significant and ongoing resistance from those who have a historical claim to the label) been redefined as an identity label and claim, based simply on the word itself and ignoring the chance in meaning, that means Woman's facilities are "labelled by ...identity" is at best an error of understanding, and at worst deliberate appropriation of a marginalised group's resources by overwriting their history with the dominant group's narrative.
The honest thing to do, if it is so important to trans women's identities that they get to be called Women, would be to relabel everything set up under the old meaning to make it clear it now applies to the new female-only group, ie the same people it always did. The fact this has not happened, and is considered so transphobic that not only are existing female-only provisions not being relabelled to stay female only but even allowing the existing provisions to become mixed sex, accepting the resulting appropriation of female resources for male and attempting only to set up new female-only provisions in addition is denounced as unacceptable, abhorrent and transphobic.
In fact, I must thank the PP for such a clear example of how redefining the existing word Woman from sex to gender identity instead of just having different words for these clearly different things can be used to justify a land grab between two concepts that are in reality different and unrelated, and therefore why the "compassionate" reluctance to name the reality of trans women's sex is so dangerous to female people's rights, protections and political voice.