The problem is, most cheaper dog foods have grain products and corn listed as the number one product. The meat products are usually way down at the bottom. It's only when you get into the more expensive, specialty dog foods is the meat higher up on the list.
I don't think there's much of an issue. I know several people who have rats and mice as pets. Both said you can give either pretty much anything as long as it isn't fried and it isn't covered in sugar. One even mentioned giving her rats a treat of baked chicken.