Maybe some sort of "Assassin" play? Each Pokémon team of six has two "assassins" and one "target", announced before the match by both players at the same time. In order to win, you must either defeat the "target" mon of your opponent (the finishing hit must be made by one of your "Assasins"), or just defeat all six in a normal style, whichever comes first.
There could be lots of fun aspects to this, such as:
-More use of moves like Roar and Mean Look to trap Assassins and Targets to kill them. Assassins would generally be hard-hitters, but frail, so trapping them to kill them would be crucial.
-Plays revolving around keeping your Target alive, such as Wish passing.