Sounds like being the adult and the most professional person in any situation feels like it has to be your job. I would also feel like I couldn't (or really wouldn't want to) talk to T about this. But the difficult friend and the cancelled appointment are two really corrosive issues that are interfering with your therapy and so you actually need to talk to T. You don't have to be the grown up here. You don't have to bite your tongue and consider anyone else's needs. It's totally okay to discuss it with T. You take care of T by paying her. That's it.

As an aside, if I were in crisis and a kind friend gave me the name of her LGBT positive T, I would be extremely grateful to my friend and recognize that move as very generous and giving. If I then didn't hit it off with that T, I would in no way disparage her to my friend--much less if the friend had asked me not to talk about her. Your friend is competing with you and it's mean. She's begging to be ex-friended.