Language is important. I think the responses would be very different if the questions were:
"Do you think that a man should be able to use the same changing rooms and lavatories as women?"
"If that same person (who was born male) now says that they feel like they are a woman should that person be able to use the same changing rooms and lavatories as women?"
Because, correct me if I'm wrong, that is a correct description of the proposal.
And I imagine the very large majority of people would say, "erm what now?" to that rephrasing.
I would have voted yes a couple of years ago because I would have thought it was transsexuals.