Even then, the dog would probably crush the poor chinchin. Or, if the chinchilla was also in a ball, send it spinning into a wall...
Only possible option would be super-sizing a chinchilla. Just keep breeding the biggest ones until they become as big as dogs. And then the only suitable name for the lil (well, actually big) chinchin would be Fluffy. You wouldn't even need a coat in winter, you could just pick up your dog-sized chinchilla and stay nice and warm.