Covering a graph with non-overlapping cliques
Thus I'm still curious if there are other more efficient approaches to solving the problem. 


If you take the complement graph $\overline{G}$, then your problem corresponds to a coloring problem.  Cover by cliques in $G$ is the same as covering by independent set in $\overline{G}$.

The problem is para-NP-hard in the unweighted case and the problem is just hard no matter what you do.

There is an algorithm computing the chromatic number in time $O^*(2^n)$ using exponential space, and in time $O(2.2461^n)$ using polynomial space.

The fastest algorithm I know to get the coloring runs in time $O^*((1+\sqrt[3] 3)^n) = O(2.4423^n)$.

[1] Fomin, F. V., &amp; Kratsch, D. (2010). Exact exponential algorithms. An EATCS Series — Texts in Theoretical Computer Science.  Springer.