You do realize that an adult medium is usually the same size as a child's large or x-large right? Well, just in case you didn't know that, my granddaughter is going from a girls 8 to a 10 and she can also wear almost every adult X-small and small she puts on. She's 9 and weighs 50 something pounds.
An adult small will have thigh high cut's in a bathing suit, room for boob's, darts, a deeper cut arm hole for comfort, a tiny bit smaller waist and a bit fuller hip/thigh, they are basically the same garment but with slight, less than 1/2 inch difference.
I don't think it sounds like your girl is really flabby type of overweight but more like she's solid. If she's flabby and has rolls of fat hanging then of course she's overweight but exercise is all that will take care of that.
It sounds like she's not getting anything that is making her this way.
If she ate a whole fried chicken for dinner every night, if she ate a couple of Big Mac's a week, etc....that would tell us she's eating foods that are contributing to her weight issue. What you are saying that she eats isn't really enough calories for a kid her age.
Are any other kids in the family this size? Do the kids run big in hubby's family and then when they hit this certain age they shoot up and get thin? I know at least one family that does that. Their little girl is a year younger than my granddaughter and she can't even wear an adult large in shorts or pants. BUT she's solid and doesn't have a roll anywhere. She is a round solid little girl. She consistently ate healthy foods like salad, veggies that were steamed from fresh produce, she didn't get candy or pop, etc....
This little girl is suddenly shooting up and slimming down. Her dad's family is like this with all the girls. She is turning into a beautiful young lady. Her diet nor the foods she did or did not eat had nothing at all to do with this either.
So I caution you that what you wrote that your daughter is eating is not really enough calories for a growing child. She sounds like she probably is really hungry at dinner.
I am saying she is just one of those kids who are going to be bigger. A meal a couple of times per month out will not make her fat.
If the school food program was not healthy every single student in the building would be fat also so don't start stressing out and making her meals to take with her. My goodness, I'd say she needs a couple more years before she needs to worry about being a fat kid. Let her enjoy being a kid and having fun. She doesn't sound like she's all that overweight unless it's rolls of fat hanging everywhere. If she's solid and muscular then she's fine.
In my granddaughters ballet class she is one of the kids that are the middle size. She has 4 girls in there and each of them are either an adult small or adult mediums. One is older and is in a size 7 girls, very skinny and short.
They are all athletes and active but each one is built differently, not one is flabby and a roly poly. So if your daughter is like these girls she is fine.