Is it possible to detect a simple negative-weight cycle of weight $N$ in polynomial time?
No, there isn't (not unless P=NP).  Take an unweighted directed graph on $n$ vertices, and set all of the edge weights to $-1$.  Now there is a simple cycle of weight $-n$ if and only if there is a Hamiltonian circuit in the original graph.  But detecting the existence of Hamiltonian circuits is NP-hard.  Therefore your problem is NP-hard, too.