I'm not sure I'd consider crossing the Alps between mid September until the start of June; between those dates you're more likely to meet bad weather on the mountain passes. If you're happy to use the tunnels, you'd get yourself another month or so each side, but I can't see there's a lot of fun in breathing in everyone else's fumes for so long. You could avoid the Alps, though, by heading on down through France and crossing into Italy at Menton. Then you'll have warm weather all the way down to Bari (even if you get a few thunderstorms). I've not ridden in Italy; people say that, outside the mountains, the straight roads make it a bit dull. Dang Romans.
Your other route is to keep further north, heading east through Germany, and then over the Carpathians - they're half the height of the Alps, so easier to cross. Aim for the hills in Germany, like the Harz mountains, and this could be great route.
Kalymnos is magical. Please don't tell anyone about it. (no, I'm not a climber).