@SarahAr
I would suggest you try reading this. It is produced by the EHRC specifically to help political parties navigate the EA2010.
www.equalityhumanrights.com/sites/default/files/equality-act-2010-guidance-for-political-parties.pdf
Pay special attention to this on p.8:
Selection arrangements that aim to improve the representation of one particular group but would unreasonably reduce the selection prospects for another protected group are likely to be disproportionate.
As PP have pointed out, the crux of this case is that we (and JJ) do not believe that self-ID transgender women have the protected characteristic of sex, hence cannot be included on AWS even under a proportionate aim to increase representation of people with the characteristic of gender reassignment (if they even have this, which it is by no means clear that they do) because it would then no longer be an AWS and thus potentially illegal.
JJ could be disadvantaged since, if it all it takes to get on an AWS is a declaration that you are a woman, this would reduce the selection prospects for women with the protected characteristic of sex, via the mechanism that has been put in place to address their existing disadvantage.
As always this is a fight about whether transgender women should be lobbying for facilities/services/opportunities that meet their needs and requirements, or taking over the ones that women have spent decades fighting to get (and by taking them over, completely undermining their usefulness for women). I expect a lot of women would support them in the first - I would. I won't support the second.