Could it be an old type press stud or snap button. Looks a bit larger than the ones i have on one of my shirts in place of buttons. They were also used, I believe, in leather work or military clothing which may have justified a more decorative finish rather than the metal or pearl studs found on some clothing.
The screw may have been used to tighten up the snap mechanism, indicating that this one was intended for long term use.
Just some thoughts.