I would second Butterflywings suggestion, but I would use staples on the dog. lol NO just kidding. Does sound like puppy needs some lessons of what is not allowed but that doesn't help your quilt.
Do you have some more of the fabric to where you can sew same over same? You can add a piece of batting and do a simple cross hatch to keep it in place. Then add a new assembled piece of four over the chewed piece. You can match the quilting by doing it from the back. I think it would be worth a try.