Is this variant of set cover NP-complete?
As quicksort comments, you can convert any set cover instance to your special case by adding dummy elements. Take a large enough pool of dummy elements, and put all of it into a new set $S_1$. Then add subsets of $S_1$ to the other sets to satisfy your cardinality condition, taking care to leave some of them out. Every solution has to take $S_1$, and otherwise it just has to be a solution to the original problem.

This is of course not a formal reduction - this is left to the reader.