It's obvious that they know they haven't got a leg to stand on, or as others have already pointed out, they would be rushing to get a legal decision as soon as possible, not wasting cash to postpone it coming to court.
On the contrary, AWS that include trans and non-trans women are clearly legal. The only issue is whether the Labour party implemented their AWS in a way that is legal - i.e. can they justify their selection criteria, did they breach party rules etc.
If the campaign wins, trans inclusive AWS are not going away. All that will happen is the Labour party will correct the procedural deficit and continue.
Litigation is always risky however strong your case. You don't say to your opponent, I will concede you have standing and allow you to go to court. What you do is argue
(1) You have no standing to bring the claim
(2) Trans inclusive short lists are legal as they are proportionate
(3) Alternatively they are legal as for the purpose of AWS, sex includes people's gender identity
(4) Even if we have broken the law, you have suffered no harm - so get no damages.
You hope point (1) wins as it deals with the litigant prior to the full trial. Otherwise you expect to win on (2). Failing that (3), is a back up but you don't expect to win on that point etc.
You also don't concede a legal question on point (3) - as you expect to the win the case on point (1) or (2).
So the campaign is wasting the Labour party's money and time, when Labour should be focussed on bringing down this government.