My “one screen” mechanic works!
My whole game takes place inside a theater. When the player tries to exit one side of the stage, however, he’s put right back on the other side.

I’m a little sad I couldn’t get a cloth sim working well in Unity, but I rendered the video out in Maya and used a video texture plane instead.
Overall I’m pretty well pleased with the effect. I might add an option to disable the video playing though. Maybe it will get too distracting later?
For now it’s time to get some sleep and start fresh in the morning!