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.
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.