King Louie's in a really weird position.
Since he wasn't in the original story, he's a purely Disney-owned character. Or at least, he would be...since his likeness was heavily based on Louie Prima, Prima's estate gets a royalty each time King Louie is used. This is why he's either removed (like in The Jungle Book 2) or replaced with an expy (House of Mouse used an expy called "his brother, King Larry" who was literally just King Louie). That being said, I think video games might be okay, since they couldn't have been mentioned at the time of the contract since they didn't exist, and that he's mentioned in a skydome in Disney Infinity, and his face is on a Power Disc for the game. It's a sticky situation, but they might be able to use him in Kingdom Hearts.
It wasn't just that he was heavily based on Louis Prima, he was voiced by him, too. He was Loius Prima, for all intents and purposes. Then a couple decades later, there came a show called TaleSpin. For all you younglings, it was (an awesome) Jungle Book spinoff, and in it, Jim Cummings did a spot-on King Louie voice, and it made Prima's estate very uncomfortable. It led to a lawsuit over negligent misrepresentation, which was settled out of court, and since then Louie has only appeared when using a different voice (such as in the (not as awesome) Jungle Cubs cartoon, or the upcoming film where he'll be voiced by Christopher Walken).
EDIT: Further on this, if I had to hazard a guess, archival footage may be perfectly okay. A year after the lawsuit over royalties and likeness ended, in 2001, Disney released a psone game called Jungle Book Rythm n Groove, and King Louie is present, but only to sing his classic song, using the exact recording from the movie. It's probable Square could put King Louie into the game if the only dialogue they use is dialogue from the film, because it doesn't count as a new performance. It would necessitate Louie's role to a
bear bare minimum, but at least he'd be present.