Alaska looks at the total distance of the itinerary, not just each individual segment. OAK–FAI via SEA clears the 2,000 mile threshold, so a paid First Class ticket qualifies.
Because SEA is a same-day layover on that qualifying First Class itinerary, you should be able to access the Alaska Lounge in SEA in both directions.
A couple things to keep in mind:
This applies to paid First Class, not upgrades
Lounge access is valid during connections, not just origin
Just scan your boarding pass in SEA, no extra steps needed
People run into confusion because the segments themselves are shorter, but Alaska doesn’t break it down that way for lounge rules.
So yes, grab a drink in SEA and enjoy the stop.