The best designed option would probably be a sort of carrier that rotates. It'd have to grab the disk by the edges, but if it could move the disk from the original orientation, to 90° for the edge pic, then 180° for the back pic, it could then rotate to like 270° to drop the disk.
That might be the most reasonable option.
It would require variable focus on the camera, since the surface it's photographing is closer for the edge picture.