Dividing a complete graph into two cliques with maximal sum of edge weights
I'm going to assume you didn't mean to end up two (maximal) cliques, but instead two disconnected complete graphs. Those are not the same, e.g. for $n = 6$ you can end up with extra edges that don't form any other maximal cliques otherwise:



If that assumption is correct, your operation is called a bisection of the graph. You want to maximize the remaining weights, and thus minimize the edge weights of the edges that are cut.

This problem is NP-complete, and you can find a reduction in "Some simplified NP-complete graph problems" by M. R. Garey, D. S. Johnson, and L. Stockmeyer in Theorem 1.3 and the preceding paragraph.