Hi, John! Your intuition is correct. For your SETUP, you want that one big comp so you can work spatially across screens.
How you actually OUTPUT your work depends on the video engineering and playback system requirements. (And those ideally depend on the creative requirements.)
If your videos will only ever sync-roll, playback might actually prefer one file at the full 1536×1536 canvas size, or pre-positioned in a 4K raster.
If they do specifically want 4 separate files, you can speed up your renders by doing the screen crops with output modules instead of comps. With 4 output comps, you have 4 render queue items, and therefore you have to render 4 times. With 1 full-size working comp in the render queue, and 4 separate output modules attached to it and handling the crops, you get all 4 outputs from just 1 render.
Depending on the playback system, they may have specific frame rate and codec requirements, too.
TL;DR, you’re off to a great start, and I think you’ll have to talk with whoever’s responsible for playback to make sure you’ll output exactly what they’ll need.