Eating disorders, bingeing, starving, over eating, are a common response to abuse. Emotional, physical, sexual. As you say it's a form of control they can have so they will cling to it.
There's also the fact that she looks a lot like her mother. Whom she has seen ballooning in front of her eyes for years. She may be reacting to that by deciding never to be fat and because she couldn't admit that to anyone much less herself, she won't even know why she's restricting her food intake without a lot of counselling. Which she will never get while she remains under the McMansion roof and Sarah's iron control.