Can you post a bigger picture of your object? Cant really see where this edge going. The best way to modify is by hand. Delete edge (what you dont like) & split polygon tool. Just make sure you delete the remaining vertices after you delete the edges, they get left behind and you end up with ngons which is a lot worse than tris and things like your insert edge loop tool will no longer work. Looks to me like where ever that edge is going it is not a very good use of geometry so if it's meant for tying something together there must be a better way of doing it in another area.