I really think they will accept your bags early. I regularly (well, 3 or 4 times per year) drop them at least three hours before the flight. The Aegean desks are all common to every flight (i.e. not a desk for each flight) and are staffed all day. I am assuming you are Gold so surely they will do everything they can to help at the Gold desk.

As somebody else said, there is a paid luggage store which is just to the left as you exit the secured area at arrivals.

There are also lockers in the A3 lounge but I'm not sure they are big enough for a large suitcase. You could try though - go to the lounge after you arrive. If nothing else you can have coffee and breakfast before heading off to the city. As already mentioned, it's before security but you'll need to show your onward boarding passes to enter the 'A' departure area and explain what you are doing.

Also, I wouldn't totally give up on getting the bags through checked. It's unlikely but still worth asking (very nicely) one more time before you leave the US.