What I think would be sort of middle ground is a satchel that can only hold Poke Balls that you have to equip on the player like its a piece of armor, so most likely in the chest slot, and would have between 10 - 15 extra slots in it.
To me, it would make the most sense if the only way you could open it outside of battle by using a key bind. You would also be able to use the Poke Balls stored inside during battle.