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Use the EXP share to level more than 6 pokemon, duh.
When you go to stomp the latest gym or whatever the hell it is now, take the 5 pokes that won't be shitting on the gym leader and replace them with 5 lonely randoms from your box. You might find a new poke you like, or even just get some "not top 6" favorites out to play. The limit of 6 on you is garbage anyways, it's a tiny ball I just fucking stick it in my backpack, could fit hundreds honestly.
Go to battle? use any amount agreed upon, perhaps limited to the lowest number of pokemon held by any one participating trainer.
IE: I am engaging in a double battle against my rival and his sidekick, with my sidekick helping me.
I have 14 pokemon.
My sidekick has 8 pokemon.
My rival has 4 pokemon
My rival's sidekick has 16 pokemon.
We all agree to use 5 pokemon each, with my rival giving us a bit of a boost since his 4 pokemon are better trained than my 14.
This is his decision, by the rules if no agreement can be made it defaults to the lowest number of pokemon held by a participant, in this case being 4.
Of course, EXP shares need to be changed to held items and made more available, perhaps reworked so that any poke carrying one counts as having been previously switched in and out again when exp is awarded, thereby dividing the exp instead of just magically making new exp.