Complexity of unbalanced bipartite isomorphism
If you are willing to relax the 2 to some other constant, then yes: this can be solved in $O(c^a)$ time for some absolute constant $c$. You can view this problem as isomorphism of hypergraphs, where the vertices in $A_i$ are the vertices of your hypergraph, and the vertices in $B_i$ are hyperedges. Luks (STOC '99) showed how to solve hypergraph isomorphism for hypregraphs on $a$ vertices in $O(c^a)$ time for some absolute constant $c$.