Find perfect matching faster than MCM, in graph that has perfect matching?
Hopcroft-Karp isn't necessarily optimal, and you can already find asymptotically faster algorithms in the relevant Wikipedia pages: https://en.wikipedia.org/wiki/Matching_(graph_theory)#In_unweighted_bipartite_graphs, https://en.wikipedia.org/wiki/Maximum_flow_problem#Solutions.  Note that you can use any network flow algorithm to find a perfect matching.

For instance, you can achieve $O(n^{2.373})$ running time using fast matrix multiplication, though this is only a theoretical result; in practice it's slower.  However, this is already enough to demonstrate that there's no way we're going to prove that you can't do better than Hopcroft-Karp (because you can).  For sparse graphs ($|E|=O(n)$), Madry's algorithm achieves $O(n^{1.43})$ running time.  Neither of these use/exploit the fact that the graph already contains a perfect matching.