You can alway move the knot down on the bungi cord to tighten it up.
Not sure you want to change out to bridle line. I see this on kite with bowed leading edges.
To change out the bungi cord use bridle line a little longer than twice as long as your bungi cord. attach the same way. Run both line through the nock, back down through the hole or fabic tab and then back over the nock. The knot will need to be where the bridle line meets the nock to lock it in place.