Favorable graph decomposition for dense graphs to solve independent set problem
The problem is that most decompositions (I know of) are themselves much harder to compute/approximate than INDEPENDENT SET.

However, if your graphs are really very dense, their complements might be sparse. Thus, it might be worth considering solving CLIQUE in the complement of the input graph. This can be done in $O(nd\cdot 1.45^d)$ time (Eppstein, Löffler, Strash), where $d$ is the degeneracy of the input graph (which is even smaller than the treewidth).