We flew back from Santorini a couple of weeks ago. BA is handled by Swissport there. I'm almost certain there was a Club check-in desk. After security there's quite a pleasant shaded roof-top bar. Before boarding, you'll need to show your passport and wait in a non-Schengen area, which is indeed quite cramped and hot. There was fast-track boarding for eligible passengers, they had sole use of the first bus. We arrived at the airport 70 minutes before departure and that was plenty of time. Hope you enjoyed Santorini!