I don't have your condition, but I do have a lot of pain from having fingers cut off in a table saw years ago and bad circulation. Best thing I've found is to wear a thin pair of leather thinsulate lined gloves first. Then I bought an oversize pair of Mittens (actually milsurp ones with fur on back) and I also buy the instant heat packs from Walmart for $1. I put a heat pack in each mitten before heading out and keep my hands inside the mittens with the thin gloves on. Then only pull my thin gloved hand out of the mitten when time to shoot. Seems to work pretty well. When processing the deer, I cannot afford to have it butchered or won't pay $60 to $100 anyway to have it done, I keep water drizzling in the sink warm to hot and stop after every few cuts to warm hands back up. I find it keeps me from cutting myself too as I have done before when really cold. Hope it gives you some ideas.
These are the like my mittens:
http://www.sportsmansguide.com/net/cb/new-us-military-n4b-flyers-mitt.aspx?a=490282These are like the gloves:
http://www.sportsmansguide.com/net/cb/carhartt-cowhide-leather-driver-gloves.aspx?a=972579These are like the warmers:
http://www.sportsmansguide.com/net/cb/grabber-weekender-6-pk-warmers.aspx?a=499081 Oh and I also put the kind on the bottom of my socks that stick to them so you can walk all day without them moving and I throw a couple in my lower back if it's really cold too or use one of those back heat stick on pads for sports injuries. I live in Michigan and it gets plenty cold here too at times. Especially, sitting for opening day.
***edit***
One other trick I use is that when I'm field dressing the deer, I actually stick my hand right down in the guts often to warm them up if they get too cold. Kinda nasty, but does work and I don't like sticking a bloody hand back into my clean gloves. I keep some of that alcohol based hand cleanser and paper towels in the back of my jeep too, so when I'm done cutting him/her up, I can clean off my hands and get the gloves back on quick.