Max-clique in line graph of hypergraph
Every graph G is a line graph of some hypergraph, for instance the one that has the edges of G as its vertices and the sets of edges neighboring each vertex of G as its hyperedges. Therefore, finding cliques in line graphs of hypergraphs can be no easier than finding cliques in arbitrary graphs.

Of course with your limitation of edge size $r$, one can only do this starting from a graph $G$ with degree $r$, and max clique in bounded degree graphs is not so hard, so this doesn't rule out the $n^{poly(r)}$ algorithm you're hoping for.