Computing unique subset intersections
Let $u$ denote the size of the union of the sets in $S$, $n\leq 2^u$ denote the size of $S$, $h\geq n$ denote the size of the output, and $S_i$ the set of sets resulting of the intersection of $i$ sets from $S$ (in this sense, $S=S_1$), maintained in lexicographical order. For all $i\in[1..n]$, $|S_i|\leq h$.

$S_2$ can be computed in time $un(n-1)/2\leq uh(h-1)/2$: each binary intersection takes at most time $u$, and there are at most $n(n-1)/2$ of them. $S_4, S_8, \ldots$ can be computed in time $uh(h-1)/2$ each in a similar way. Other $S_j$ which are not power of two can be obtained by combining those results, up to $S_n$.

The total running time would be  within $O(unh^2)$?