You should be able to get a good caribou hunt with two caribou for $1,200 to $1,400. The trick is flying to the city directly. There is no need to come into Anchorage. However, you will not be able to do everything for $2,000. Even with air fare and air taxi coming in at $1700 -$1800, you still need licenses, food, extra baggage charges, plus you have to get your meat home. Expect to pay another $500 if you donate most of the meat.

There is no place you can do a fly-in for moose as a nonresident for $2,000. You could try to hunt off the road system, but it is crowded, and you will be competing wth locals who know where the legal bulls are.