Or how about this:
The Japanese decided to keep Jeju after Imjin War, providing them with a constant supply of horse; and then Toyotomi Hideyori won the Japanese civil war, deciding to realize his father's dream of conquering Korea and ultimately China, this time with Spanish help.
Japan never controlled Jeju Island during the war itself, and even if it had, it would have made virtually no sense on a logistic level, as Tsushima (which was the main naval base during hostilities) was geographically much closer, not to mention that it was much easier to invade Gyeongsang than Jeolla, as the latter was extremely mountainous. The currents also flowed in the opposite direction, hindering the invaders from consistently suppling troops to the western island. Additionally, the horses present on Jeju itself would not have provided a significant impetus for a direct invasion, as they were of a smaller breed, and would have been unable to sustain a large force due to their low number.
Also, Toyotomi Hideyori's position after his father's death was extremely unstable precisely because of the devastating failure of the Imjin War, after which Tokugawa Ieyasu established a competing faction, as his core forces had not participated in the invasion of Korea. The Japanese court also decided to resume cordial relations with the Korean one soon after the conclusion of the war, indicating that it was willing to adopt the diplomatic route after extensively exhausting the military option. In addition, the fact that various daimyo had openly opposed the Imjin War, along with their rapid switch to Ieyasu soon after the conclusion of hostilities, suggest that even if Hideyori had won out, his power base would have been much more marginalized in comparison with that of his father. Additionally, Toyotomi Hideyoshi had issued several orders to curb growing Christian influence since 1587 due to various cultural tensions (and in anticipation of military ones), along with more suppressions and an outright ban in 1614 under Tokugawa Ieyasu.
As a result, it would have been extremely unlikely, if not impossible, for Hideyori to suddenly change tact and ally with Spain if there was a looming threat of a Spanish invasion of the archipelago soon after the defeat in Korea and another civil war within Japan. On the other hand although Joseon had severely underestimated Japan before 1592, leading to lax preparation, it would not have made the same mistake twice if another invasion had been in the works soon after 1600, as it had increased its number of diplomatic missions. As a result, Korea would have rapidly shifted its focus to militarization, butterflying the political strife that had occurred IOTL, in addition to a much more significant mobilization by China, and a potential counter-invasion of Japan, which would only reconfirm the
status quo after devastating the four countries involved in the second conflict.