Finding a 4-clique among $k$ node groups
First of all, the greedy strategy you outlined does not work. It may be possible that the node with highest degree is in a clique with lower total degree than the clique with the maximum degree.

Second, this problem is clearly in P because we can simply check all $\binom{n}{4} = O(n^4)$ cliques, checking they meet the partition requirement, and keeping track of the one with largest total degree. It's possible there's a more efficient algorithm though, especially considering this naive one does not use the partition information to rule out any cases.