I knew an old german shepard who lived in an avocado grove where my band used to practice. He was a fat happy dog who ate mostly avos and gophers which he could catch quite well. I watched him eat one once, no fuss, just crunch crunch crunch and it was gone.
I don't know that the fur would hurt any, but I'm sure it's not digestible. If anything keeping the skin on would give a little more fat. Too bad it's so hard to pluck a mouse.