Has this bipartite graph problem been studied?
This can be solved in polynomial time: it is a system of linear equations over the integers, so you can solve it in polynomial time by forming the Hermite normal form.  This might not be the most efficient way to solve it...

If you want to express it as a flow, you could try the standard transformation of replacing every vertex $v$ with two vertices $v_\text{in},v_\text{out}$ with an edge $v_\text{in} \to v_\text{out}$; edges directed towards $v$ are replaced with edges to $v_\text{in}$, and edges out of $v$ are replaced with edges out of $v_\text{out}$.  I don't know whether that will help in your situation.