Is it a good idea to expose yourself to situations that trigger fear and anxiety and use meditation to get through it [?]
Maybe -- and it depends on what you mean by "meditation". But since the context is about CBT, I'm not sure this would be advised, specially without professional assistance.
or is the desire to overcome fear and anxiety "bad" because it's technically a desire?
Desires for well being, for good, and further aspirations are not considered "bad", on the contrary. In Buddhism, two technical terms for things we strive to overcome are craving (pali taṇhā) and clinging (pali upādāna). These are specific kinds of desire that present problems.
(And fear and anxiety are only temporary states anyway and should be recognized for being just that.)
Indeed, they should. But again, in context of CBT, I don't think recognizing the temporariness of such things are usually enough for remission.
Would it even be possible to meditate through a triggering situation since one wouldn't be calm enough to see clearly in such a situation?
Again, it depends on what you mean by "meditation". If you mean bringing up attention and mindfulness, calming the breath and "observing", yes, naturally.
Of course, the success will mostly depend on one's skill, and how strong these fears and anxieties are. Thus, if the former is weak and the later are strong, intentionally putting oneself in such situations where more harm can be done is probably a bad idea.
To put it more simply: could meditation be used as a substitute for CBT?
As someone who is not a doctor, but only familiar with DBT (a form of CBT) and who was close to people with disorders who relied on such therapy: hardly. While "mindfulness meditation" is considered an important and integral practice in the process, I personally think is nowhere near an adequate substitute of the entire therapy program. I also don't know of any research suggesting the remission efficacy of disorders with meditation only vs. CBT (or DBT).
If not, does Buddhism encourage or discourage CBT?
I don't see anything in these therapies that would present a dilemma with the Buddha's teachings.