I didn't mean to accuse you of anything, was just working off limited info so apologies if it came across this way!
I wouldn't bother with Nurses / GPs, not to take away from the work they do, but I don't think they have a lot of training in disordered eating and nutrition beyond the very basics.
Can you speak to a therapist? There are psychologists that specialize in nutrition. Or a nutritionist who works with people with eating disorders.
If your daughter is binge eating that does fall under disordered eating already. This could be related to emotional eating, stress (at school or other), and lots of other reasons. Or it could be that her friends all do it so it's a habit.
Is she getting enough sleep/rest? If she's got lots of activities on and school maybe reaching for sugar is a quick fix for being tired?
I think stopping allowance now would come across as punishment, unless you're able to really make it seem like it's unrelated.
This is just general thoughts so you might be or not be doing this already, but I think you need to make sure you're not labelling food as good/bad in front of her, commenting on food, on her body etc. Is she confident in herself otherwise?
I would sit down with her and give her a safe space to talk. How is school? Is she struggling with anything? Does she need extra support? Don't judge or comment & just listen so she can properly open up if possible.
For the actual discussion about food, I would tell her that she's beautiful either way and that her weight is the least important thing about her. Ask her if she'd change anything about the meals/food at home and if she's happy with her activities etc.
Building up that trust now will help you progress these conversations to a point where she will see you as someone who doesn't judge & who she can trust and that will make it a lot easier to be more direct over time.
Finally, when she tells you things, please don't be defensive as she will close off again. You might not be but I'd imagine it can be hard sometimes when a teen is criticizing the way you do things, but it will give you a lot of insight on the deeper reasons why, which really is the first step.