Another (a bit risky) way is for me to book two separate tickets: CDG-SEA ($800) and then SEA-FAI ($450).
Yield management systems are indeed crazy. I can't imagine why they'd put a $1000 premium on booking the direct CDG-FAI itenary.
The $700 CD-SEA ticket is indeed a bit bizarre:
CDG-FRA on Lufthansa
FRA-LAS on Condor
LAS-SEA on Alaska