I was about to wade into the discussion when I realized this thread was started back in October, so I'm hoping whichever way you decided to go, it has worked out.
Anyhoo,
I've never heard of CAT before, but I know of other attempts to combine psychoanalysis with CBT. Schema Therapy is one example of this kind of thing. It's not entirely a woo idea, but it's not entirely un-woo either.
The problem with CBT is that it tends to work well with people that have severe problems like panic attacks and phobias. It tends to work less well with people that have long term mild to moderate depression.
It's easier to change your false beliefs, and irrational ideas, about something when the beliefs are really quite irrational, and are causing you significant difficulty. I think the idea of the psychoanalysis part is to look at the long term effects of the thoughts or behaviours that are causing problems.
Although I've seen research that backs up the conclusion that CBT works less well on people who are mild to moderately depressed, I haven't seen any to suggest that combining it with more traditional psychoanalysis works any better.
I've been reading through this thread, and the therapist-doesn't-talk-about-himself-at-all thing sounds really, really woo.