Minimum cost max flow network problem with an alternative flow cost
This is sometimes called the minimum edge-cost flow problem or fixed-cost flow problem. As you suspected, it is indeed NP-hard, even when the network is bipartite. It is listed as problem ND32 in the list of NP-hard problems by Garey and Johnson: 

M.R. Garey, D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York, 1979. 

An approximation algorithm is discussed here: 

S.O. Krumke, H. Noltemeier, S. Schwarz, H.-C. Wirth, R. Ravi. Flow Improvement and Network Flows with Fixed Costs. In Operations Research Proceedings 1998: Selected Papers of the International Conference on Operations Research Zurich, p. 158--167, Springer, Berlin, 1998. 

The idea is to solve a standard linear-cost flow problem in which edge capacities are unchanged, but the cost of one unit of flow along edge $e$ is defined to be the edge weight divided by the edge capacity. This will give a solution that may weigh up to $F$ times the weight of the optimal solution, where $F$ is the flow value; but it will not weigh more than that. 

If the approximate solution is not good enough, you might want to resort to an ILP solver.