From my experience you don't have to actually KO with the specified character for these type of missions. You just have to win with the specified character and then KO your opponent (with anyone).
Step 1: Hit with Grudj
Step 2 (and 3): Stall with Defeat: - X opp life, Defea: + X life and damage reducers.
Step 3 (or 4) Finish with appropriate amount of damage from any card.
If a mission says ko with this character you don't actually have to use him to complete the ko. You can win a round with him and KO with a different character and still advance the mission.