Motivating the Korean side to pull such an adventure would be the primary problem, and without the willingness to go through the trouble, any invasion might just end in a peace treaty without outright conquest. One important factor that limited Korea's option was the typhoon season in late summer which is sometimes extended to autumn, and this danger was decisive in ending the aforementioned 1419 invasion - Korean troops had to withdraw before typhoons could hit the island, lest they would be stranded facing the wrath of sea and hostile local people. Ultimately, Korea came to adapt the diplomatic approach over the militarist approach, and this posture continued even after the bloody Imjin war.
As noted by other posters, this is contrast to what was happening in north. Korea in against its northern neighbors was very proactive, and particularly the Joseon dynasty was willing to assert its influence over the Jurchens, regularly sent out expeditions over the Tumen to raid disobedient tribes, and fought off Jurchen (counter-)raiders, repeating for hundreds of years before the rise of the Qing dynasty. Even after the Manchu subjugation, the ambition lingered on in the form of the Gando dispute, only ending in 1909 by a treaty between Japan and Qing. Korea was a protectorate of Japan, by then.
Perhaps some external factors could be introduced to redirect Korea's attention. An early, permanent Chinese subjugation of the Jurchens probably would deter Korea from its northward ambition, or maybe have a Korea-friendly Jurchen tribe to gain an upper hand over the others so that Korean interest is secured. Supposing the northern border to be settled and stabilized by the late 15th century, if the Japanese piracy not only continue to flourish but actually grow up out into an independent force that is out of reach from Kyoto while posing as a threat to the Korean state, which it wasn't during the historical 15th century, now then maybe Korea could see permanent occupation as a solution to the never-ending struggle against the Wokou.
If the landing in Kyushu succeeds, presumably after the conquest of the Tsushima islands I would imagine the occupation to be limited to ports and adjacent coastal areas, with administrative autonomy given to the local lords, in a similar way to how the Three Ports (Sampo) were administrated during its existence. If Korean troops march off further to east, given the complete collapse of Japan's central authority during this time (the late 15th century) Korean troops would be fighting local Daimyos on the way not a centrally organized defense, and seizing Kyoto would not have been completely impossible, but how would Japanese Daimyos would react to the news of the Korean invasion I have no idea, and former pirates in Kyushu certainly aren't going to stay low forever...
