I have used bungees for training but as side reins when lunging. I don't ride in them. I like them in the fact that if they are loose enough, they seem more forgiving than actual side reins for teaching a green horse how to balance and build up a topline.
Have you simply tried to talk to her yet before you hit the pavement and find another horse? She could be trying to get a quick fix in order for you to feel like you are riding a "dressage horse" that is perfectly in frame. I would explain to her that you aren't really comfortable in riding with the bungees and can you try it without. Use yourself as an excuse like "it feels funny" or "you want to learn how to get him going without the bungess". If she says no then you move on. This is an opportunity for you to teach her. If she is open or not is up to her.