I'm in the last year of a PMHNP program and have worked in mental health for 4+ years. My clinical placements include inpatient psych and running a CBT group. I know quite a few Psych NPs who focus primarily on providing therapy (not only CBT but a variety of approaches). One of my classmates is in a private practice that combines med management & therapy.
Sure, PMHNP training doesn't go in-depth into psychotherapy. That doesn't change the fact that PMHNP can technically bill for therapy. Among various therapeutic models, CBT is one of the therapies that, IMO, is easier to pick up on. I do encourage getting more training and there are plenty of training out there for CBT.
Personally I have no interest in providing psychotherapy, but in responding to OP's question about whether it's possible to conduct therapy (CBT in particular) as a PMHNP, my response is yes.