What's complexity of this set problem which looks like "Linear Programming"?
If I'm not mistaken, the case you specify (with $k =2$, $k_i = 3$) is in P. 

Note that the condition $|S \cup S'| \le 3$ is equivalent to saying that $S, S'$ intersect (since $k=2$). Thus, if you think of the $S_i$ as vertices, then the constraints specify a graph $G$, and your problem reduces to finding a graph $H$ such that $G$ is the line graph of $H$. 

Verifying that $G$ is indeed a line graph and then finding $H$ can be done in linear time.