You may be correct. I 'know' - via an online forum - rather a lot of Lab activists. They are, without exception, opposed to women's sex-based rights. One, who didn't get involved in the arguments over the SC judgment, recently announced that he 'like most Labour members' opposed the SC ruling, but (as a believer in democracy and the rule of law,v unlike...) would seek to change the EA.
My heart sank just a bit further, as I had hoped Lab might slowly return to sanity (LDs or GINOs are a lost cause for the foreseeable).