Is there a direct/natural reduction to count non-bipartite perfect matchings using the permanent?
I would say that a "simple" reduction to bipartite matching is highly unlikely. Firstly, it would give an algorithm for finding a perfect matching in a general graph using the Hungarian method. Hence, the reduction should contain all the complexity of the Edmond's blossom algorithm. 
Secondly, it will give a compact LP for perfect matching polytope and hence the reduction should not be symmetric (which are ruled out by a result of Yannakakis) and inherently very complicated.