An old real Hessian sack was the best, train your pup to be put in it and tie it around their neck so they can't get out. They will be warm and dry in minutes.
My sprocker got hyperthermic one sleeting day. I had to carry her back when her legs were going. I abandoned the gun and put the forend in the game bag and lugged Tia to the car. Then I put her in the sack. Once home, gun collected leaving the place, she was red hot and feeling much better.
Sadly, finding a real Hessian sack is near impossible now.