In Co-op mode, Kai doesn't change much, however, his hyper is slightly tweaked, by always enabling him to attack first and his recovery rises.
Kai, by default, will perform well as an attacker, given his already buffed attack stat which will be further buffed by the attacker role. His decent HP pool should enable the player to last quite a few combats as well, before his inevitable KO. Unfortunately, Kai will only be able to take advantage of half his hyper as an attacker, as he will always attack first as a result of the role, rendering that part of the hyper useless. The ability to not face a boss's counterattack can be extremely helpful, however, especially if many of the player's allies are currently KO'd.
Kai can work decently well as a guardian thanks to both his hyper and above average HP pool. With 5 HP, he will be able to provide a decent number of shields to allies, and by using his hyper the player can avoid being KO'd by the boss until they can either recover or until the rest of the player's allies have recovered. Even better, the player will also receive an additional defensive boost by using the role and thus increase Kai's survivability.
Supporter is not a great matchup for Kai, as he doesn't have any heavily increased stats on his own, and he will not receive any attack or defense buff as a supporter. While the role does provide a +1 evasion boost, it will not increase the player's survivability much during the late game, and the player will suffer from his lack of damage output.
While viable, avenger is not an ideal role for Kai as a result of his hyper. Given that the point of the role is to receive damage to deal more damage to the boss, it is simply not a very good mix with a hyper that makes the player move first in combat and not receive any damage. That said, Kai can still be a decent avenger as a result of his HP stat, but his lack of a defensive buff of any kind may cost him heavily during a battle depending on the players' RNG.