I LOVE the idea of a dog shaped applique. I'd find fabric that matched the coloring of the dog to use. Dog bones over the smaller spots would be great. And, even if there was a label on the quilt originally, I'd put another one on explaining the dog appliques.
I know it's disappoionting/discouraging to have something you've labored over be damaged. at this point, I'd celebrate it and make the repairs a prominent part of the quilt.