That's rough, man. You're very much in a bit of a predicament, and I feel for you. Alas, here's where my lack of experience and professional ability rears it's head, and I'm hoping someone who does have more knowledge/experience can step in with some suggestions, because I'm stumped. But maybe, we can try and break it down into it's key elements and work from there.

You have a fetish, of which you're ashamed, and which you struggle to control actions surrounding it.

You struggle with restraining your own desires during moments of intimacy, and this gets in the way or ruins the moment. This causes you shame, for your failure in restraint, and then shame for having the desire at all.

Your girlfriend does not share your fetish and is even repulsed by it, due to her own struggles in life and how they've shaped her perception of it, and because she feels like you care more about the fetish than her. She's given you an ultimatum demanding you stop having this fetish or she's ending the relationship.

Here's the harsh reality: there are elements in this over which you have no control. You have a fetish. Fetishes do not go away, much less are they willed away. For lack of a better description, they're hardwired into us (near as we can tell), we don't yet understand how or why, and any "remedies" I've heard of out there share more resemblance with North Korean re-education camps than actually helpful therapies. This is a part of you that you will have to come to terms with and accept before you'll be able to make real headway with managing it. This is the good news, though. Fetishes definitely can be managed, just like any desire or body function. If I may make a suggestion, something that I have found immensely helpful for me so far in my own struggles is trying to practice mindful awareness. It's difficult to explain, but I feel it might be helpful for you to incorporate some of the practices into your life to help you manage your fetish. The Meaning of Mindfulness and Awareness - How to Work with the Mind


But, there's another element you don't control (nor should you). Your girlfriend's reception of it. And I feel this is the most difficult element in all this. You can't make her share your fetish, much less enjoy it, and from the sounds of it, it will be a loooong time and much effort on her part in working through her own struggles before she could even tolerate it. And understandably so. If I'm reading between the lines correctly, she has struggled with and continues to struggle with her body weight and her self image as a result of that. She has developed an extremely negative view of obesity and especially any perception of herself being obese, so any reminder of that (like your fetish) brings up those negative feelings inside her. And on this front, I'm at a loss for helpful ideas or suggestions. Because these are problems that you can't address yourself. You can support her in her struggles (something one should always do for their relationships), help her work through them, but at the end of the day they're her struggles to fight through. You can suppress your fetish, for a while, to try and shield your relationship from her ultimatum. But it will always crop up again, in some way or another, and if she can't accept it... I don't have the capability to see how this would work out. I'm sorry, but this is my uneducated view of it.

I know this is not helpful right now, but you need someone far more knowledgeable to help you make sense of this and help you find a path forward. And if your relationship is to survive, your girlfriend needs to take an active part in that. But know this: I don't think you're wrong for wanting to keep your relationship, for trying to save it. So far in this discussion, all we've discussed is the big problem you're having in it, but this is not all relationships are made of, and I salute you for desiring to fix this problem for the salvation of the good in your relationship.