Directed graph with bounded in-deg can be partitioned in a balanced way
This is a special version of the Beck-Fiala theorem.
Define a set system on the vertices whose sets are the out-neighborhoods of the vertices.
The in-degree condition will give that every element is in at most $n$ sets.
The theorem states that in this case the elements can be colored with red and blue such that the difference of the red and blue elements in any set is at most $2n-1$.
This means that in your digraph the out-degree of each vertex $v$ in the corresponding subgraph is between $d(v)/2-n$ and $d(v)/2+n$, so for your specific question $c(n)\ge 6n$ is sufficient.