I came across a presentation by Ryan O'Donnell regarding invariance principles. After proving the Berry-Esseen theorem, there is a slide that discusses extensions of the theorem and one that is mentioned there is a so-called "derandomized version'':
If $X_{1},\ldots,X_{m}$ $C$-nice (that is, has bounded third moment), 3-wise independent., then $X_{1}+\ldots+X_{m}$ is $O(C)$-nice.
I am not sure whether the above is a statement regarding the third moment of the sum of 3-wise independent random variables, or there is indeed some variant of the Berry-Esseen theorem in the case of bounded independence.
Inspecting the proof, I see how 3-wise comes into play, however I could not find any source that discusses bounded-independence variants of this theorem. Are there any?