Jo, my heart goes out to you. You are walking in a minefield.

A teenage girl eating other people's food if she can get her hands on it, getting heavier, and screaming that she doesn't want to talk about it: carb addiction absolutely, but keep your eyes and ears open in case there's more going on. It's possible she is self-medicating because something has happened that is stressing her: overwhelmed at school or work, or something/someone has made her feel she's not pretty or "worthy", for example. The society we live in is cruel to fat teenage girls, so it's doubtful that she is "happy being overweight", but if she's serious, she may be making herself unattractive on purpose to avoid unwanted attention.

It's odd that she has an amazing role model/coach in you (she only has to look at your own transformation) but refuses your help. If there's something driving the overeating beyond the sugar/insulin/sugar rollercoaster, it may take another teenager to talk to her to find out what it is. If you have a good relationship with any of her friends they might be able to offer some clues.

For now, there's not much you can do but drop the subject, make certain she knows you love her and that nothing she says or does will change that, then wait for her to come to you in her own time, when she's ready.

This situation may be nothing more than the growing pains of a headstrong teenager, that will sort itself out in time. But you seem like an action person who doesn't enjoy sitting around waiting, so for your own peace of mind you might benefit from seeing a counselor to hear what an experienced professional has to say on the subject. This would be a personal quest for knowledge for YOU, and not something your daughter even needs to know about.