I’m not accepting your assertion as fact. There are currently multiple civil rights movements facing organized conservative resistance.
You can invent whatever moralizing obligation you want, but I don’t think it’s reasonable expect an organized effort from the {$outgroup} community to defend the rights rights of the people who designated them the outgroup.
Existence of such groups is not your original assertion. You stated:
> People are disinclined to support your rights if they see you as advocating against theirs.
By that argument, the Libertarians would be loved by everybody, but they aren't, because it's no longer a discussion on ethics, it's a fight over holly dogma. The ACLU will no longer support people on the Right in the name of Free Speech. Lines in the sand have been dug by the left, and they no longer crosses them. The right, (ie. the MSM's fascist) however do it all the time because they are the most open ones. If you are a gay 1sh amendment absolutist, you aren't gay, you're just a nazi. The ancients Greeks are no longer the bases of our culture/civilization, they're esclavagist.
You can invent whatever moralizing obligation you want, but I don’t think it’s reasonable expect an organized effort from the {$outgroup} community to defend the rights rights of the people who designated them the outgroup.