Thanks for the tip; I will try it. While I originally tried to simply do the entire thing in Boris Red, I abandoned that attempt due to the extra steps and timeline required to edit in Boris.
So what I ended up doing was copying the original video clip, using the crop tool (with stretch to fill frame disabled) to create a rectangle around the black monitor area, and adding a Sony Chroma Key set to 20,20,20. Then I added a timeline below that, with the monitor image resized and aligned with the monitor. Below that, obviously, was the original video clip. This seems to work decently, although Sony’s not-so-quality chroma key isn’t exactly as nice as the one native to Red.