(SAWE)I would also ask her to find me some other appointment time, when that other fellow wouldn't be there.
(LG) ..........one of my bigger problems is me avoiding situations and stress.. wouldn't this be basically the same?

No, because you are not there to deal with other clients.
LG it's not as if this was a co-worker or a neighbor who is creating a problem. You are there at that office at that time of day ONLY to work with yr T, and it's very demanding work that requires concentration and focus, and safety and privacy, and this third person is affecting that.

Problems can have many solutions; just because a solution is non-confrontational doesn't mean it is avoidance.