Although I like the answer by Sourav Gosh it does not adress you problems with the women's support group itself.
As I (now, thats why its edited) understand you do not see the neccesity of a Women support group. Do you know to the full extend how your peers feel about it? Maybe it is different in other departments? It might be rather different than what you are feeling. That would be an appropiate topic for such a Women Support group and than maybe together you can decide that it is no longer neccesary to have one. So I do not think you should try to stop the group. I do think that you could try to change the focus or topics of the group.
So about the topics/sessions which are now part of the Womens Support Group I agree with the answer given by Sourav Gosh.
So to answer your question:
Are either of these even worth doing, or is my best bet to keep my head down while looking for a company more aligned with my personal values?
Edited after comments by OP: Flip genders and it indeed becomes problematic. Therefore in a perfect world/company the current is also problematic. That is why I think the topics are important. If the topics are about the problems women face (which are gender related) within the company then the world/company is not perfect yet and the group has a function. If it only discusses topics interesting to all then the world is apparently already perfect and the group should be opened to all.
So in the first case: Do not try to shut down the group. In the latter case however you can, maybe via peers or direct supervisors (or a thrust person if your company has one) let your concerns be known