I think it's fair enough for a charity that is trying to encourage a section of society to use its service to require front of house representatives to follow the marketing/branding involved, including wearing badges etc.
Obviously with volunteers there is a greater need for organizations to appreciate they need buy in from those representatives. But if a few volunteers don't buy into the organizations direction, it isn't unreasonable for the organization to say "That's very sad, but we're going ahead anyway. You're welcome to continue volunteering away from the public eye where you won't disrupt the branding/marketing."
There is of course a separate question of whether the marketing of this property to the LGBTQ community is appropriate. But it seems reasonable to me.
I don't think YABU to sympathize with volunteers who find themselves no longer able to support the direction of an organization they've put a lot of time into. But it is in the nature of large organizations that they must adapt to survive, and because of that there will inevitably be some losses along these lines. So the while sympathetic, I dont think there is anything that should be done about it.