Update: I was able to book 2 Business Class seats last night.
-LAX-SEA-ICN on Asiana
-ICN-BKK on Thai Airways
-BKK-DPS on Thai Airways (spending 6 nights)
-DPS-BKK on Thai Airways (spending 6 nights)
-BKK-LAX-SEA on Thai Airways
Overall, I was fairly pleased to get all segments in Business class. I may try to tweak it later on and see if I can get LAX-SIN-DPS on Singapore instead.