I've been conceptualizing the idea of a contest rework lately.
Here are the overall rules, with further explanations bellow. (5 rules out of 10)

1-Each player has three Pokémon, they can switch them in and out at any time during the battle without losing a turn, like in rotation battles.
2-An Attention (ATN) bar replaces the HP bar. The ATN bar is shared by the whole party and starts at zero. Pokemon can fill it up by using attacking moves.
3-There is a public pool of ATN points that all players compete for at the start of the battle. When this pool runs out, players must get ATN by targeting their moves at other players' Pokémon.
4-Mood determines the effectiveness of moves, not the defending pokemon's typing. There are five different mood types: Beauty, Cool, Cute, Clever, and Tough. Moves which match the mood are boosted, while those that go against it are diminished.
5-Each player also has a hype counter. This counter is filled by using moves which are compatible with the moods, while those incompatible lower it, neutral moves do nothing to it. When five compatible moves have been used the last one receives a x4 super effective boost.