So when a therapist demonstrated to a patient that some of their beliefs were incompatible or their arguments were contradictory, the patient might assert that the therapist was the one who had the irrational concern or obsession.

Labeling this as projection seems overbroad. If you and I are arguing and it's pretty clear you get my position/premises, and I assume that I'm totally rational, then you must be irrational. Totally valid argument, though of course the premises may be false.

If I believe non-sentient things are acting "irrationally," i.e. they should be acting the way I want, but they aren't, then "projection" seems legitimate. But it seems wrong to call it "projecting" my irrationality by believing those who disagree with me to be irrational. After all, my reasoning is (potentially) valid, it just might be unsound.