If we set up a camera to take lots of pictures (say, 20 FPS during the daylight giving us 20*60*60*15 = 1080000 or just over a million images) looking towards the north star during the daytime, could we process them on the fly to cancel out the sky glow (enormous but random) and extract the light from the circumpolar stars (faint but constant)?

We need to do it in a properly centered rotating frame of reference. We could test it by doing the night time to find the center and then (making sure not to move the camera)...