I think that members of a biological sex group are well within their rights to request areas that are specific to that biological sex. Biological females have the right to organise around those characteristics.
An organisation that thinks merely making such a request or supporting the right for those who do, is “hate speech” or deserving the label “transphobe” or that invoke an antisemitism comparison is overreacting to a extent that, yes, will probably be damaging to their reputation in the long run.
Stonewall have done great things in the past but they appear to be painting themselves into a bit of an ideological corner here.