OK, first off if your vet said to cut the hair very short, your vet knows better than anyone what your pup needs.
I have a rescue 'mutt' name Tramp. He is a wonderful old soul and sweet as can be but he is very cold natured. I live in Florida and sometimes he gets cold in the summer from the A/C and wants to wear his sweaters ALL winter long. Please remember, hot air rises, cool air falls. The lower to the ground he is the colder he gets.
That being said, you've gotten some really good ideas for coats. For Tramp ... he 'prefers' the polar fleece. In addition to having a sweater on his 'daddy' (my DH) has a chair in our bedroom for Tramp (to keep him off the floor) and covers him up when they 'turn in for the night'. We keep a piece of polar fleece on our bed for our other rescue, Benji, to sleep on so he doesn't get jealous!