I think you would need to get the reflections in the same pass. It’s basically ray tracing in real life: emit light from a bunch of points (screen pixels), “calculate” how they bounce through the room (ie adjust the brightness of the corresponding photo), and combine them all into a single image.
I think this would be a final additive pass after the pixel setup. Would also allow you to turn it off selectively...
The more I think about it the more I want to try it out.