The Kingdom of Sarawak gets rich from gutta-percha harvesting and resource extraction. They modernize their navy, takes it on a test rip to Korea, and impresses the state enough for it to give away Jeju Island. In exchange, Sarawak would give (or sell) Korea antimony, oil, coal, rubber, and other materials for industrialization.
Or have Sultan Abu Bakar of Johor tour Korea along with China and Japan (he did this IOTL) and accepts the island as a protectorate in exchange for Johorean investment.
Or have King Chulalongkorn of Siam go to Korea during his Grand Tour. Repeat.
Hey, you did say "Give Jeju Is. to the most unlikely country".
Or have Sultan Abu Bakar of Johor tour Korea along with China and Japan (he did this IOTL) and accepts the island as a protectorate in exchange for Johorean investment.
Or have King Chulalongkorn of Siam go to Korea during his Grand Tour. Repeat.
Hey, you did say "Give Jeju Is. to the most unlikely country".