How hard is it to count the number of local optima for a problem in PLS?
Consider the maximum matching problem in bipartite graphs. The family of feasible solutions consists of all matchings, and local search is performed via finding augmenting paths. The problem belongs to PLS since an augmenting path can be found in polynomial time if a current matching is not maximum, and maximality can be checked in polynomial time. Any local optimum is a maximum matching (i.e., global optimum). However, it is #P-hard to compute the number of maximum matchings in a bipartite graph.

Since a local optimum can be found in polynomial time, the problem is unlikely to be PLS-complete. So, I'm afraid this is not an intended answer (your question restricts to PLS-complete problems). However, I should point out that counting the number of local optima can be hard even though one local optimum can be found efficiently.