You could limit the user to using a script to duplicate the record:
Set Variable [$ID ; Value: YourTable::AutoEnteredSerial ]
Set Field [YourTable::DuplicateOf ; $ID ]
If you have FileMaker Advanced, you can set up a custom menu where selecting Duplicate Record from the Records menu or pressing its keyboard shortcut performs this script instead of the standard duplicate record action.
Thanks Phil. I like the idea of the custom menu. I'll explore going down that path.