Is this a question of trying to turn a national trust property into "the one with the gay owner" and the volunteers feeling this is a limited focus? Sorry it seems that there is more to the story and I am struggling to read between the lines.
I am fully supportive of rights for all -my own sexual orientation is not conventional. However I feel that sexuality is personal (NB personal and private but not "secret"). I don't understand why an organisation that looks after old houses is asking volunteers to publicly "support" the fact that people have different sexual orientations. It's a fact of life, it's fully protected by law, accepted by the vast majority, with a minority that are fully aware that their views are unacceptable.
I work in the public sector so I'm in receipt of public money, too and would be quite miffed to be asked to wear badges to support whatever the latest campaign is. Wearing badges is childish imo.