How to approximate minimum clique edge cover
Here is a paper that deals with this problem:

John M. Ennis, Charles M. Fayle, and Daniel M. Ennis: Assignment-minimum clique coverings. Journal of Experimental Algorithmics 17(1), 2012.

It also gives some heuristics and experimental results. They don't give worst-case approximation ratios; minimizing the number of cliques instead of the sum of their sizes has only fairly weak approximation guarantees, so I guess it would be similar here.