Here is some general answers to the best of my knowledge:
The USFS has cabins in south central in decent black bear areas and there are some moose around at least some of those cabins. That would be Chugach National Forest. You can't drive to any. Either walk or fly in. Horses are used to get to a few.
The USFS cabins in southeast (Tongass National Forest) are in deer and bear areas but no moose. Most of those are on lakes are require fly in. None are drive in. As best I know, all the USFS cabins are reserved by internet, not telephone or in person.
The US Fish and Wildlife Service has cabins as well and many of those are in decent moose and bear areas. They are listed by refuse so cabin for Kenai National Wildlife Refuge are listed under their site. Other Refuges have cabins but you have to look them up by refuge, not the agency overall as best I know. Keep in mind that their cabins are reserved by telephone or in person only, not thru internet.
Alaska State Parks and the National Park Service has cabins as well. No hunting in actual National Parks (except some subsistenece) but hunting is allowed on some other land managed by the NPS such as areas around Yakatat.
Most of the cabins realistically require fly in which is possible. All the goods ones fill up quickly during hunting seasons so you have to plan quite a a bit out, both for the cabin and the charter flight in and out.
Remember traveling with black powder is really not an option so you will have to make sure anything you need is available wherever you plan to charter out from. No problem out of Anchorage or Fairbanks getting BP substitute, maybe OK in Juneau, nearly impossible elsewhere in the state unless you know someone who lives there and has it barged in.
Hopefully that helps.
