I thought this was a rpg for your PCs to play while they're having a down time. Is there any rpg that do that?
Yep! There's a few in the indie that are explicitly designed to be played during other games. At the moment, I cannot remember their names, but I've definitely seen some. And there's also a few (specifically fishing ttrpgs) that slot into other campaigns pretty easily even if they're not intentionally built to interact with other systems.