This is an interesting idea, and one that I'm not sure I fully agree with.
Now before reading what I have to say, please understand I support you and your choices in regards to your body, and I am not trying to convince you to change anything about yourself. I'm just commenting on this particular idea, that fatphobia is genocidal in nature as I think it's inaccurate and you can combat fatphobia without trying to call it genocide.
You say, the "foundational belief that fat people shouldn't exist".
I don't think that's a really accurate description of their beliefs. If a fat person loses weight, do they stop existing? No of course not. They are the same person. Just like if a thin person gains weight, they don't "stop existing".
So I don't think you can apply rhetoric of genocide to diet culture. Especially given that a lot of diet culture is related to "health", all the stuff about the "obesity epidemic" is about trying to SAVE LIVES or extend lives, ie keep fat people here with us for longer. Rather than trying to make fat people "not exist".
Sure they might not want "fatness" to exist. But is fatness truly such a fundamental part of who you are, that you will disappear if that part of you changes?
For that matter, is "thinness" such a fundamental part of your dieting friends, that if that changes they will disappear? Maybe they think it is. But that belief is false. The self will remain, regardless
Look, I used to have an eating disorder, and absolutely that eating disorder was based on fatphobia as it related to myself. But more generally it was about my own relationship to my body. The feeling that I could control my body as if it was a separate "thing". The belief that my image of myself, my concept of myself, was the real me, and that I needed to control that concept to feel ok. It was tied to an "overcontrolled" nature in general. Trying to control my behaviour and how I was perceived, so that I would be loved. And that was tied to the idea that if I changed in a way that wasn't my choice, something vital about myself would be lost.
I've now recovered. (I wrote a whole series about how I recovered on my blog).
But part of my recovery was decoupling the "image"/idea of myself from my actual self, which is first person consciousness of life.
I don't think your dieting friends want you to "not exist". But I think they think maybe that if they gain weight, something about themselves will cease to exist.