Gosh, tough to tell. A neighbour's Lab had a front leg pinned in mlutiple places after he was shot, chasing chickens. Stupid people still let him run loose and the first time I met him he was charging at me through lots of snow and turned out to be friendly and skiied 5 km with me. I met him several more times and the cold did not seem to bother him.
Two human neighbours have plates, one in skull, one in face, and both feel the cold badly.
Good luck. Dogs are like little kids and don't usually show discomfort till it is really bad. What did your Vet suggest?
Seems like a good question to me.