>>48273165Outside rewards would be cool. Something like unlocking seals for pokeballs, more TMs available for sale at certain spots if you have more pokemon caught, have battle facilities give you BP rewards if you have more pokemon too.
Could also do it indirectly by rewarding people for using different pokemon in their playthroughs or at challenges. Like, say you have your gym card where you keep the badges and they also register which pokemon you used in each gym. The more unique pokemon you use in different gyms, the more rewards you could get after beating the gym. instead of getting just 1 TM, if you beat the gym with a team that's at least 3 new pokemon, you could get another TM or early access to a new area. I think this would be the best way to do it personally.
Also something like having a side or daily quest where an NPC will challenge you to a battle but give you 3 random pokemon you MUST include on your team to proceed. Give him a bigger chance to list pokemon you haven't caught and there you go.
And FINALLY, how about some more trades in the game? Have some active NPCs say something like they'll trade you a trade evolution pokemon if you catch and bring a certain pokemon for them or something.