Complexity of Finding Optimal Synergistic Set Packings
This is an instance of the standard network flow problem called "Project Selection", and hence can be solved efficiently (even in practice). See (what's currently) Section 24.6 of Jeff Erickson's lecture notes for a nice explanation. It's covered in most introductions to network flow in good undergraduate algorithms courses, so if you don't like Jeff E's notes, I'm sure you can find plenty of alternatives.

The general idea of project selection is that you have a collection of projects, each with a cost/benefit (positive or negative) and a collection of dependencies, and the goal is to find the set of projects that (1) satisfies dependencies, and (2) maximizes the benefit of all the selected projects.

For your case, you would have one project for each set $S_i$, and also one project for each $e_j$. Each $e_j$ project has no dependencies, and each $S_i$ will depend on the $e_j$ in $S_i$. The benefit of $S_i$ is its payoff, and the 'benefit' of each $e_j$ is the negative of its cost. You can see that reasonable ways of selecting projects correspond to all the ways of picking solutions to your problem, and the objective values are the same on either side of the correspondence. 

I'll just add that you're really only using a special case of project selection, where there are "good" projects that only depend on "bad" projects that have no further dependencies. Project selection doesn't assume this structure, and can be solved on an arbitrary dependency graph. You might find this useful in your application where you've got multiple phases of computation.