Update: success!

I went back to the airport and visited the checkin counter. Someone escorted me upstairs to the Qatar office and about 5 minutes later someone emerged with my lost item. It was all rather painless.

There’s little chance I could have retrieved it without someone from the checkin counter escorting me. The “office” is a nondescript, locked door that is clearly not intended for customer service.

Kudos to the Qatar Airways people at SEA. They were friendly and efficient in helping me with this.