I'm super excited that this is being talked about. There's obviously a lot of things to consider but I wanted to post ,in my excitement, about PokéBalls
all of this is coming from one with basic knowledge of Pokémon and decent enough knowledge in open legends to play while looking specifics up occasionally.
One could use an advantage system to an agility roll based on static and conditional bonuses of the ball used, one could use the gen 1 system of just getting a random number in a parameter based on the power of the ball, catch rate of the Pokémon, and status effects, etc..
To fully consider this, though, we'd need to know how challenge rating for catching each Pokémon is determined.
A lot of this could be just simplified by the will of the GM, but I'd appreciate any ideas.
Edit: There may be some confusion but what I would like optimally is some system that allows for me to consistently compensate for when someone uses a Pokeball vs. when someone uses Taming or Influence.