This is a tough one... you are an adult and have your own responsibilties to attend to (namely your doggie!) but you are still under your mom's roof and therefore under her thumb
Until you move out and get your own place, this will just be a losing battle.
If I were you, I'd find another dog food or else your mom is just going to keep going against your wishes. Find one that doesn't make her sick and that is still good for your doggie. Make it easy on her so she'll actually feed her properly.
Unfortunately, your doggie is YOUR responsibility and perhaps you should have considered your living situation before buying a pup. Until then, just keep mom happy so she will keep the dog healthy, kwim?