Can I make a File Share Room with the ‘Named Organizer’ Model?

No, the ‘Named Organizer’ licensing will not allow you to make a room that can be started by a ‘Presenter’, even with the ‘Auto-Promote to Presenter’ enabled.
A workaround is to just record the meeting room and make the recording link available…make it say 30 seconds of recording and that would be ample time, they don’t even need to click ‘pause’ as the end of the meeting will still show the active ‘File Share Pod’:
This would also not count against any licenses and can be viewed by hundreds of people at a time.
Just a thought to get around the Named Organizer licensing model in this case