I'm not sure I understand your classification scheme, though. Transgender isn't a gender. It's just a description for people whose gender identity does not align with their birth sex. Classifying by male, female, transgender would be like classifying by male, female, and Catholic.
Yes, I anticipated this and should have addressed it the first time.
For whatever reason, a very large number of people are uncomfortable sharing a bathroom with people of a different biological sex. To reiterate, I don't share that view, but it's commonly held. I agree with HF that the male/female breakdown doesn't work well once you recognize that male <> man and female <> woman. But the sex breakdown is still relevant to most people even if it no longer aligns with the way we talk about gender. So we still allow a bathroom for females and a bathroom for males, because that's how our current social privacy concerns break out. For people who don't feel that they comfortably fit into either of those assignments, we provide a third alternative.
It seems to me that this sort of arrangement does the least violence to anybody, given current social preferences. If you want to tell me that male/female thing is stupid, I will agree with you. No problem. But the overwhelming majority of folks who support men/women restrooms I think really mean male/female, and if there's no special reason to respect one but not the other. The important thing is to provide accommodation for folks who don't fit neatly.