I am not sure how to proceed with the proof. That is, I’m trying to prove the contapostive, but I don’t know how to prove that a composite number greater than 3 will not be in either set. Insight would be appreciated.
" I don’t know how to prove that a composite number greater than 3 will not be in either set. "
That isn't the contrapositive and it isn't true.
The contra positive of: If $p$ is prime and $p > 3$ $\implies$ $p\in_6$ or $p\in _6$. then the contra positive is
$p\not \in _6$ and $p\not \in _6$ $\implies$ $p$ is not prime or $p \le 3$.
So prove that if $p\in _6, _6, _6, _6$ then eithe $p$ is composite of $p \le 3$.
Might be worth noting. If $m \in [k]_6$ then $\gcd(k,6)|m$. Do you see why that would be true?