Think of it more like "IRL raytracing", where a ray (the beam) is cast and the result for a single pixel from the point of view is captured, and then it is repeated millions of times.
Even if you had a clock and camera for every pixel, the sync is dependent on the path of the signal taken. Even if you sent a signal along every possible route and had a clock for each route for each pixel (a dizzingly large number) it still isn't clear that this would represent a single inertial frame. As I understand it even if you used quantum entanglement for sync, the path of the measurement would still be an issue. I suggest not thinking about this at all, it seems like an effective way to go mad https://arxiv.org/pdf/gr-qc/0202031
E: Do not trust my math under any circumstances but I believe the number of signal paths would be something like 10^873,555? That's a disgustingly large number. This would reveal whether the system is in a single inertial frame (consistency around loops), but it does not automatically imply a single inertial frame. It's easy to forget that the earth, galaxy, etc are also still rotating while this happens.