Packing sets to maximize overlap
When $k=1$, the problem is trivial: the optimal solution is to put each set in its own configuration.

When $k=2$, the problem can be solved in polynomial time using graph matching.  Build a complete undirected graph with one vertex per set.  The cost of the edge $(S_i,S_j)$ is the cost of putting $S$ and $T$ together in the same configuration (i.e., the weight of $S_i \cup S_j$).  Now the optimal solution to your problem is given by the minimum-cost perfect matching in this graph.  That matching can be computed in polynomial time using standard algorithms.

When $k=3$, the problem is NP-hard.  In fact, it is NP-hard to determine whether a graph can be covered by a disjoint union of cycles of length $\le 3$ (i.e., whether you can find a set of cycles, where each cycle has length 2 or 3, where no two cycles have any vertex in common, and where every vertex appears in one of these cycles).  This implies that your problem is intractible when $k\ge 3$.

The proof of the hardness result I alluded to can be found in Section 3 of the following paper:

Clearing Algorithms for Barter Exchange Markets: Enabling Nationwide Kidney Exchanges.  David J Abraham, Avrim Blum, Tuomas Sandholm. EC 2007.

That said, not all hope is lost.  There are techniques available for finding a solution that will be close to optimal, using integer linear programming.  For instance, you can express your problem as an instance of integer linear programming and then apply an off-the-shelf ILP solver.  See the paper above for some more sophisticated techniques that might be applicable in your situation as well.