One has to have priorities. Once the basics are taken care of, there might be a few other things that you should do if you insist on shooting in really cold weather. Most of this will be old hat to duck hunters, but some of it is sporting clays specific.
Warm coats tend to be bulky.
Bulky coats make it hard to mount the gun. You might try wearing your heavy coat over your vest and sweater. You can remove the coat for the few minutes it takes you to shoot a station. Most of the better clubs provide coat racks at the stations for just this purpose. Same thing with gloves.