By rights, you should countersink the frame, and probably the wood behind it, but the metal is so thin that localized dimpling is not likely to be an issue.
Choose a small enough flat-head screw that the head diameter just grabs a bit more than the OD of the hole in the frame.
Choose a small enough flat-head screw that the head diameter just grabs a bit more than the OD of the hole in the frame.