You should tell the vet the collar is too big...Coco should definitely be able to walk with it on. If they don't have a smaller one, you could try trimming it down some. As long as his mouth and tongue can't get past the edge, it will work to keep him from licking his sore spot. Dogs never like those collars, but tough.

They get used to them. You're the mom and you know best, right?