Can I just say that you are taking the easy option. You are thinking short term peace not long term good.
Giving in and giving her the rubbish now to make her happy is not helping her.
I am sorry that that is very blunt, but I was fat growing up (I am fat now!). It is a horrible existance being a fat child - it is miserable beyond belief being a fat teenager. And having been fat all your life, it is a milion times harder to change.
Break it when you are young, learn the healthy eating patterns young, cos trying to change a lifetime of poor eating habits when you are an adult is damn hard!
My children are not fat now, nor will they be for as long as I have any control over it. I will NOT put my kids thru what I went thru.
You are not being kind to your daughter by allowing this to continue.
She will scream, yell, cry, say she hates you - she will probably mean it for a while. But if you don't change now and for good, then you risk condemning her to a miserable, lonely and probably seriously bullied time of it.
I was bullied, mocked, had few friends. When I got older, I was the one in the frumpy gear while my friends had the short skirts and the pretty tops. I was the one who never had a boyfriend - teenage boys are very shallow!
Your daughter needs to you do what is right for her, not what will make for an easy life in the here and now. No child is going to choose healthy food over the sweet/stodgy stuff! - that is what parents are for - to make the sensible choices for us before we are able to make them for ourselves!
Aside from anything else, there are all the health problems - kids are getting type 2 diabetes because of their weight! Not to mention heart problems etc!
At school pe was a nightmare - I was the kid who needed the excercise the most, yet I was the one who got out of it as much as possible to avoid the humiliation.
And don't say that she is popular, she has lots of friends etc etc - cos my parents would have said the same about me - if it is true at the moment then I tell you now it wont last, she WILL be bullied, she WILL be miserable and she WILL get trapped in the I hate my weight I will eat crap to cheer myself up I hate my weight cycle. When that food goes down your throat, yes, you feel happy. But then you feel guilty, horrible and you wish you hadn't had it.
Give her the healthy menu and ignore ignore ignore the outbursts. Make her see you mean it - she eats what is in front of her or she doesn't eat. She will eat! She is playing a waiting game with you because she knows you will give in and let her have the stuff she prefers!
It won't be easy. It will be very hard indeed. But you can't avoid doing the right thing, just because it is difficult. She will not thank you for this when she is a 16 stone teenager who get laughed at every time she leaves the house and who none of the boys are interested in. - that might seem like nothing to us as adults - but I remember how much it mattered at the time, and that the experiences of the teenage years linger for life!!
Take the control while you can. Let her be mad. Let her be hungry. But for God's sake don't let her stay fat.