Min cost set of edges to connect 2 subgraphs s.t dist of nodes between subgraphs <= K
When the costs are all equal, you are looking for the fewest number of edges to bring your graph to being of at most diameter $k$. This is the graph diameter augmentation problem. You are looking for the edge-weighted version, which I believe has been studied as well.

There are a number of versions of diameter augmentation out there: some that fix or bound the amount of bridge you can build and the task is the bring the diameter down to as small as possible. Or, your version, you have some $K$ in mind and you want to find the cheapest amount of bridges in order to acheive that $K$.

Naming of these problems is non-standard or inconsistent in the literature, but searching for 'graph diameter augmentation' will get you on your way.