A camp doctor would probably work okay (Jon Pertwee dressed a little camply so the aesthetic of a camp doctor has already been proven to work) but I wouldn't recommend him showing off his sexuality as kids would probably be a little grossed out (remember young children do this kind of thing plus it would be PC for the sake of PC). But yeah a camp doctor, in fact I would be completely behind that, it would be funny if nothing else and add a great splash of humour to the series without having to resort to other methods of pointless humour like dinosaurs on spaceships.
I wouldn't see the point of a transgender doctor, you could argue if the doctor had a female incarnation then that would be transgender, but I am against the idea of a female doctor strongly because yet again it would be PC for the sake of PC, I couldn't envision it working, there would be endless jokes and it really would kill the show or at least cause another early regeneration. The majority if not all the people I have talked about this with in real life agree that a female doctor would be dreadful and it would be pointless to change a successful format. After all, there is a huge difference between changing the age of the actor playing the doctor and changing the gender of the doctor. It would be an unnecessary risk and it would be near impossible or even completely impossible to pull off right. I mean you remember the backlash from the younger fanbase about the doctor being older now, but just imagine a gender change. I don't want to sound ignorant or anything, but it just wouldn't work.