PN guy said people were more supportive of gay rights than transgender rights. If he bothered to engage his brain, he might realise that's because gay rights don't take rights away from other people, so we can all support them without losing out ourselves.
What does it mean to support or oppose transgender rights? Transgender people should have human rights, the same as everyone else. Opposing the right of transwomen to have access to women's spaces is not a denial of rights to transgender people, it is upholding women's right to a single-sex space. Transwomen can campaign for their own spaces. Supporting the right of transwomen to be treated as women in all circumstances is a denial of women's right to single-sex spaces.
Framing this as a transgender rights issue is incorrect. Nobody is advocating the removal of rights for transgender people. The transgender lobbyists are, however, advocating the removal of rights from women. This is a women's rights issue.