I am wondering if a C# script can be uploaded during game play as it is being played on a mobile device as in some sort of script reader.
There should be possibilities to create some kind of runtime compiler but I doubt you need to go that far (And I can’t assure Unity will support it but I think I read something unofcifial about this long ago).
Maybe best for you is AssetBundle, it allows to get a full game object with script on. The problem is that the script is isolated, it may know about the existing ones in the app, but the other way around won’t do.
Best solution is an update of your app, at least it is safe and fairly common.