He's a Mandalorian, all Kun needs to do is make it galactic knowledge, if Mandalore doesn't accept his Mandalorians will look at him as weak, it's a win/win for Kun, If Mandalore accepts, he's already won the Kaggath, if he doesn't, the Mandalorians lose faith in their leader.
^This
Mandalore has to fight Kun if he challenges him. If they fight, Kun wins with ease. If Mandalore refuses, his soldiers lose their respect and faith in him as a leader. Now the Mandalorians are fighting each other to decide a new leader. Kun takes advantage of this and systematically wipes out the Crusaders, just like he was doing with the Jedi.