How about stove pellets? I figure you'd have super easy access to them since you're in an area with snow. They are made from sawdust (pine or oak) are fairly cheap (around $4/bag here) and they will break down to sawdust as soon as they get wet, but the really help kill the odor (of course you already know that). Or can you get some sawdust from a mill or woodworker near you?
Myabe use a combination of things - lime, sawdust/stove pellets, shavings/straw?
Down here we use sawdust in various forms for bedding for everything from horses to ferrets. Very absorbent, gets warm when stored in piles (we keep piles of it outdoors down here - it works even if it gets rained on) and keeps odors down.
It's a little slow to compost, but worth it when it's finished.