It won't hurt the pelt (providing you don't spoil them unthawing) but it is a pain. As a teenager I skinned bobcats for half a dozen guys or more. Some of them would bring them to me froze whole, or occasionally if I got too many at the end of the weekend I would freeze some whole and thaw them and skin them after school as I had time. If I never have to skin another frozen cat, I'll be happy. Hang them head down to thaw, and skin them as soon as you can, usually when they are still partially frozen between the front legs, to avoid green belly. But it is far preferable to skin them first and freeze the hide.
Smaller animals like mink and marten are not as big a deal, and are often frozen in the trap when you get them. Again thaw them without a lot of heat and skin them when they are thawed. I once was selling beaver to a guy who was paying better for them on the carcass frozen whole than I could get for put up hides (he wanted the carcass as well as the hide) but the last few I had he flaked out on me and didnt buy. They didnt skin bad, but you freeze your hands skinning a mostly thawed beaver.
A lot of northern guys deal with skinning frozen animals all the time, because they are frozen when they get them, ask them how many of the carnivores have green belly. That is just something you have to deal with when using lethal sets in that weather, but no need to deal with frozen critters if you don't have to.