Complexity of finding weighted edge-disjunctive triangles in a graph
Here's a reduction from edge coloring that I believe shows this to be NP-complete.

Let G,k be an instance of edge coloring: that is, we wish to know whether the edges of graph G can be colored with k colors. Make all edges in G negative, and add k new vertices to G, each connected by positive edges to every existing vertex in G.

Then, if G has an edge coloring, the modified graph has a set of edge-disjoint valid triangles that includes all of the edges in G: simply form a triangle connecting each edge in G to the new vertex that corresponds to its color. Conversely, if there exists a set of edge-disjoint valid triangles that includes all of the edges in G, then one can use the identities of the new vertices in each triangle as colors in an edge coloring of G.