The question we were asking is on what basis do they pick the men they admit to a women’s institution? If it’s “only men who are trans” that’s unlawful discrimination, because it’s “because of” the protected characteristic of gender reassignment.
the example in the EA guidance as to what is meant by “exceptionally” in the context of admission of the opposite sex is the son of a member of staff to a girl’s school. Being the child of a staff member is not a protected characteristic, it’s a lawful way to pick the exceptions to the “girl’s only” rule. I doubt that applies to “women and exceptionally, trans-identifying men” criteria. But - it has never been tested.