Hi Tom,
I haven't been to see a CBT therapist, but I've been delving pretty deeply into Stoicism over the last eight years, which is what CBT is explicitly based on, in addition to having practiced various kinds of meditation for the last ~40 years. I recently read
The Philosophy of Cognitive-Behavioral Therapy: Stoic Philosophy as Rational and Cognitive Psychotherapy by Donald Robertson, which goes pretty heavily into the connection.
In general, I've found Stoic practices to be incredibly powerful additions to vipassana meditation, and ones which blend pretty seamlessly into the Buddhist approach. I see both of them (and therefore also CBT) as something like distant cousins, both being great-great-grandchildren of whatever kind of embryonic analytical intellectual tendencies were present in the Indo-European culture that they both descended from, before the proto-Greek and proto-Indo-Iranian groups split off from each other about 5000 years ago. And there are always suggestions and hints of cross-pollination between India, the Middle East and the Mediterranean region from the first millenia BC and earlier, so there might be several waves of later influence in addition to their cultural roots.
Theravada Buddhism (at least, to the extent I understand it) and practices like CBT that are based on Stoic practices are really complementary, because in a sense they are like mirror images of each other. Both insight meditation and Stoic exercises are cognitive and analytic practices that often involve deconstructing unskillful assumptions about our experiences, and both are embedded in a moral, ethical and metaphysical framework that help you make practical decisions about how to live in the world. But Stoicism lacks any of the sense of transcendence that you find in Buddhism; although things and events that occur in the world are seen as fleeting and temporary, the Stoics don't really see them as illusory as the Buddhists tend to, so there is less of that tension that so many people practicing Buddhism often express - "Why does it matter if I do anything in the world (go to a doctor, work on a career, etc) if it is all an illusion?"
Buddhism, on the other hand, takes the basic perceptual insights of Stoicism and really pushes them all the way to their logical limit. If you do Stoic contemplative practices, and I'm assuming the same is true of CBT, you normally push them as far as you need to to avoid or overcome the cognitive or emotional problem you are having, but not much past that. Getting good at that kind of contemplative practice for its own sake isn't really a thing for the Stoics in the way it can be in some forms of Buddhism, as far as I can tell. But doing more intensive Buddhist insight or concentration practices does seem to supercharge the effects you can get from things like Stoic (and I'm assuming CBT) exercises.
I think practices that are rooted in things like Stoicism fill an otherwise unmet need for the Western Buddhists who are attracted to it, the need to have the effects of the practice be unapologetically geared towards functioning in the relative world instead of geared towards trying to transcend it.
Josh