Pike's Lodge and Alpine Lodge are both next to the airport. I routinely stay at Pike's; they have a shuttle and are very friendly.

As for getting to the airport, there are three AS flights all leaving about that same time...one to SCC/BRW, one to ANC, and one to SEA. I routinely fly in and out of FAI and there is rarely a long security line. Getting there by 6:30 should give you plenty of time.