That's a tough one, besides recording/streaming mods related to stuff to state someone is, or mods like CMDCam.
I think the closet people have to what your looking for with another person in your world but not pausing the game isn't really a thing.
You have other players in the game for multiplayer/LAN but your game can't pause since it's one of those host/player situations and not really what your looking for. Or there is screen viewing like Discord's. But having say another camera to view things or another player to view things live would probably be two players, but one is streaming the 'scenario' while the other does their own thing I guess. That's what I get from what you have described and similarities to such a thing/situation (it's what comes to mind at least).
But a 'mod/program that allows for viewing things like live and having another in your world and it won't pause when you work in your own client' is a cool idea but no clue honestly if one exists besides things I can think of doing other such things/having other purposes. What you have stated would be a cool idea though if someone knew how to develop it to suit such a need.
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'Niche Community Content Finder' for fun and support of creators. I look for mods that interest me or need a bit of acknowledgement or bug fix testing/reporting. I also make modpacks/maps
Creator/Maintainer of Fabric, Rift & Tesla Mod Discussion Threads/Niche Mod Youtuber.