All best wishes to MyrtleLion, hope you're feeling better soon!
Re logical fallacies in the respondents' supplementary, the first one that strikes me forcefully is the one about needing to demonstrate that transwomen are statistically just like other men. Para 10:
But more fundamentally, to be successful in the indirect discrimination claim, Mrs Peggie would require the Tribunal to find as a fact that trans women are, in all respects and for all purposes, indistinguishable from cis men. [...] Again, the protected characteristic of gender reassignment simply would not exist if as a matter of fact trans women and cis men were wholly indistinguishable.
Having based some piece of law on statistical differences between men and women we do not need to reargue it for every definable subset of men. Suppose, for example, Fife had allowed a gay man, not a transwoman, into the changing room and things had proceeded as before, until in the end the argument had said instead:
But more fundamentally, to be successful in the indirect discrimination claim, Mrs Peggie would require the Tribunal to find as a fact that gay men are, in all respects and for all purposes, indistinguishable from straight men. [...] Again, the protected characteristic of sexual orientation simply would not exist if as a matter of fact gay men and straight men were wholly indistinguishable.
I don't think anyone would be tempted by that argument, but I think the two are logically parallel. There's an attempt in para 8 to suggest that it's somehow important that a transwoman "presents, to the outside world, as a woman" - but no attempt to connect that into the argument and explain what difference it makes.