Hmmm...dunno about it fitting into the story exactly. It could just be that after each boss fight, the player gains a badge or becomes stronger as a whole. By that I mean that their Pokemon gain bonuses while fighting. Attack, Defense, Sp. Attack, Sp. Defense, Speed, etc are increased depending on which boss has been defeated. It can be just for Pokemon that aren't Shadow Pokemon, which might be best to be honest. For instance; Mirror B. once defeated could grow the player's bonds with his Pokemon so that their Speed/Defense increases a certain amount when in battle.(Like the badges in the games boost a Pokemon's strength's)
So each time a boss is defeated, the players bonds with Pokemon increase and make their non-Shadow Pokemon stronger when in battle. Something could be done once the player has defeated all of the bosses. Like a special move could be gained as an HM that they can teach all of their Pokemon if they wanted to. They just can't be a Shadow Pokemon to learn it, for two reasons: Shadow Pokemon can't learn TM's/HM's and they're unable to really be taught in their tainted state anyway.