I think the anxiety is the bigger issue as it could well cycle out of control and she's already self harming (scratching herself). This is only likely to get worse as she goes through her teens. Are the school aware of her anxiety? Have you tried to find out what makes her anxious? Ideally she should have a mentor or be able to see a counsellor.
Controlling the food intake of a teen is next to impossible since they have access, I assume, to unhealthy food at school and on the journey there and back. I wouldn't blame her dad for giving her a card to use, as he is right, she does need to fit in and lots of teens buy unhealthy drinks and snacks so it becomes normal for them.
I also don't think a bit of exercise would tackle the problem either, though it would be good for her general health and possibly her mental health too. But you'd need to find something she likes doing or find a positive reason for her to, say, walk to the shop with you (to choose food for dinner?).
It also sounds like food has always been a big issue for her, and this is not new. Maybe the weight has piled on as her growth has slowed down, and it may improve if she has a growth spurt.
I think talking to her calmly and pointing out what is and isn't healthy is good, but nagging isn't going to work. My eldest is early 20s and somewhat overweight. It started around 15 when she changed schools and had to take a bus from the town centre where there were plenty of shops. (Even if we gave her no money, she would shoplift back then).
She would now like to be slimmer and has recently started eating better, rather than constantly buying takeaways and junk food. She also walks to work (30 minutes) most of the time. She has made these decisions for herself as she could easily get a bus and has the money to fund takeaways. Whether it will lead to weight loss remains to be seen.
I don't think there are any easy fixes and a determined teen will find a way to subvert your attempts, although I still think you should keep modelling healthy eating and make sure you either don't have any highly fatty, sugary foods in the house, or keep them locked away.