Still on my NC here, but:
It's a bit of a misnomer to refer to "left" and "right" here - mainly because it leads to the confusion that's come up several times already on this thread, i.e. the economic vs. the social spectrum of positions in politics. Here, we're clearly looking at a "social conservatism" vs. "social progression" axis though. They are, of course, not entirely independent variables. As in: it makes sense that someone who supports ideals of equality would inevitably realise that their vision is not particularly compatible with unfettered capitalism - and vice versa.
Still, it's an important distinction to make in this particular context, because what's actually happening is an internal rift in the socially progressive camp with both sides' stances also being partially shared by social conservatives:
GC feminists and conservatives will generally agree that biological differences between men and women are real and matter - but this tends to be viewed in the context of it being a source of oppression by feminists and as a normative statement by conservatives ...
... who also very much tend to believe, however, that gender is very real and matters. And should be part of determining the manner in which an individual's life is lived. And this, again, is obviously a position very much shared by TRAs (but not by GC feminists). This is very much also the reason why social conservatives are by no means unified on trans issues either. If you look at countried like Iran, I think you'll find it rather hard to argue that their trans policies are informed by abundant wokeness - rather, they're just so dead-set on ideas on gender that they're happier to throw biology under the bus if push comes to shove.
As for the question in the OP: my personal litmus test for working with the "dark" side is usually women's reproductive issues and homosexuality. Hence, I've generally no problem collaborating with the most rabidly neoliberal "women in business" outlet imaginable - but any organisation associated in any way shape or form with "pray the gay away", foetal heartbeat arguments and the like are out.