This is a sad situation. I want you to realize that beneath the issue of whether you should stay or go is another, more pressing intimacy problem. Many couples are afraid to become too close, and they create and maintain distance in various ways. For some, distancing can occur through an affair, alcohol abuse, working late, fighting, etc. In the case of your relationship, it sounds like weight gain is the distancing tool. And, it sure works, because every time your girlfriend gains weight, you run for the hills.
These distancing moves are totally unconscious. It's not like your girlfriend consciously forms a plan to pull away. So, first you both need recognize that beneath the smokescreen of the weight problem lies a mutual fear to emotionally connect. (If you recall, there were many break-ups in this relationship even before the weight became an issue. So, I think the fear of closeness is an old, longstanding issue for you both.) I also think your girlfriend gains weight when she is not receiving what she needs in the relationship. She eats to fill a void. Eating for her is also a symptom of emotional emptiness, a symptom of something which is lacking in the relationship.
So, staying or leaving is hardly the point. You could break up and be in the same place with your next girlfriend--scared to connect and periodically pulling away. So, you must decide, do you want to take off, start over with someone else and end up in this same place--alienated from your lover and not sure if you want to stay or go; or do you want to stay and work the problem out right now. If you decide to work these issues out with your girlfriend, then read on.
The first step is to face your feelings for each other. Often the fear of closeness covers a fear of facing the feelings that closeness brings. There are lots of feelings and fears that are not being said directly in your relationship. So, you both must sit down and truly talk with each other. Instead of dealing with the weight issue head on, I suggest that you don't discuss the eating directly, but instead talk about the feelings which motivate the eating. Ask her: What does she feel toward you when she eats. (I think she eats when she feels angry, and since she may be afraid to telly you directly, it feels safer to swallow her anger.)
Also ask: what does she need from you that you aren't giving her. Beneath the fear of intimacy lies the fear of directly confronting your anger. I think both of you detach rather than directly discuss what's pissing you off. And, you will both need to face your feelings of anger and talk about them if you want a relationship that works. Read my 10 Tips which can help you constructively discuss your angry feelings.
And, finally, you will need to talk about what is scary for you both about being close. If she weren't heavy, and you had no roadblocks to intimacy, how would you both feel? I have given you a start. If you need help in continuing your conversations, you can call me, and we can have a three-way conversation.