Is it possible to detect a simple negative-weight cycle of weight $N$ in polynomial time? Given a directed graph and an integer $N$, is it possible to detect a simple negative-weight cycle whose edges sum to $N$ in polynomial time? I thought about modifying the Floyd-Warshall algorithm to check if the diagonals equal $N$ as they get set, but I realized this wouldn't work if a vertex appeared in multiple negative-weight cycles.
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.