Perhaps removing from the watch and try stretching and pulling with lots of force from underneath and side to side. A dull crochet hook or needle from a local sewing craft store might be able to pull it back?
Appears you might also be able to cut the end of each snag and tie together in a square knot?
Consider chatting with Apple Support and sending them the photo of the band and your "CUTE" puppy; who knows, you may be chatting with a dog lover who has a sympathetic ear. The answer is always no unless you ask the question.
Get Support – Apple
https://getsupport.apple.com/
Call Customer Support (800) MY–APPLE (800–692–7753)
or online https://getsupport.apple.com/
or call AppleCare Support at 1-800-APLCARE (800-275-2273)
Outside the USA—Contact Apple for support and service by phone
See a list of Apple phone numbers around the world.
https://support.apple.com/en-us/HT201232