Meta-x = x about x. E.g. meta-theory is a theory about theories, not a theory about itself.

https://en.wikipedia.org/wiki/Meta

(Douglas Hofstdadter made a career of cheerfully pointing out that a theory about theories must include itself as the kind of theories it is about -- "I am a strange loop" and all that. But still, what the language means is that.)

Right but that doesn't apply here. Meta almost means "higher-level" as you're using it, but emotional processing isn't a higher-level cognitive skill. It's just a different type of cognition.

Meta implies a hierarchy.

I see the problem here. It's too strong to say that being good in using your emotions (say, as an actor) is a form of level 1-cognitive grasp of a level-2 cognitive process.

"Non-epistemic" might be an alternative to "non-cognitive".

GNU/cognitive skills

(but really, the term "cognitive" used in this context boils down to g-loading, so "non-kernel processing" is as apt as anything else — an improvement over "non-cognitive" at the very least.)