Crown Prince Sohyeon's survival would be a decent start, I think. He was apparently open to Western practices, like medicine and Catholicism, and met with Jesuit missionaries in Beijing while a hostage of the Qing Empire. This openness was also apparently led to his death, as his father, King Injo, was very opposed to that sort of mindset and Prince Sohyeon died of mysterious causes after his return to Korea.
Kill Injo early and you get a Jesuit-friendly king willing to introduce Western practices into the peninsula before isolationism can be cemented as a policy for the next couple centuries.
Also, Jeju Island's history with the mainland has been...troubled, to say the least. It was long a place to send political dissidents and raise horses and has a history of rebellions and poor treatment by the mainland, so that being leased wouldn't be unthinkable of, imo.
I say best scenario could be Korea being pressured by Ming or other neighbors and Portuguese able to trade weapons and help defend. Thus establishing a good relationship and this could develop into influence, trade and missionaries. Help if the Portuguese fmfont join the Iberian union.
Korea at this time and for most of its history was extremely isolationist, even more so than Japan, so consider that the Portuguese may have to use force against Korea or gunboat diplomacy a la Matthew Perry.
The Joseon Kingdom wasn't particularly willing to open up even when attacked by neighbors (even the devastation caused by the Japanese in the Imjin War and the Manchu in both Manchu invasions of Korea over 50 years didn't see the Joseon allow foreigners in) or by imperialists trying to open it up (the Joseon violently resisted attempts by both the French and Americans before Japan finally forced them open).
Military and economic pressures didn't get the Joseon to open up OTL, so I wouldn't assume that more of either would necessarily mean success (could even reinforce isolationism, like the French and American punitive expeditions did). Military and trade matters were traditionally looked down upon by the Joseon court so it'd take a significant shift in attitude in the court to get either to hold enough appeal for major policy change.