I moved here to Idaho from Fairbanks.
Hunted caribou many times there, but the hunts close to Fbks are controlled use (40 mile herd) and usually meets the quota fairly quickly, usually not long after season opener (Aug 10th.)
You might want to consider a bush plane to get to other areas outside of the controlled use area etc.
Try Rick Gold, he's a great pilot and could get you on bou--upper Sacha River, Upper Good pastor etc.
Also, if you are in shape the Haul road has incredible bou hunting, but you need to bow hunt them by the road, or hike in 5 miles to use a rifle. You'd need to reant a vehicle for that action.
Wife and I did this 5 mile hike in hunt for kicks in '05. It was tough but memorable. About 13 miles total.
Left a helluva rack out there on the Tundra. We were hauling all of our gear and meat from a trophy bull in one trip.
Can't eat antlers...have a house full anyway.