I know you love the dog, but you do not like EVERYTHING that he does. I recently repaired a quilt that had a large burn hole. I agree with BUTTERFLY. Make a duplicate square and carefully hand stich it over the orginal. Turn the quilt over and cut a piece of batting that matches the size and shape of the hole. Now cut a rectangular piece of backing that is larger than the hole. Turn edges under 1/4 inch. Press and, using a basting stitch, stitch all around the backing patch. Using a two sided fusible (I believe that it is called WunderUnder), cut a piece the size and shape of the batting. Fuse to the batting, using a towel so that the batting doesn't melt to you iron. Remove the the paper from the WunderUnder and fuse to the backing patch. Hand stitch around the backing patch and then remove the basting stitches. The patch on the backing will be barely visible and the front will be like new. QUESTIONS?? PM me