Thanks guys. I did try the KLM website and it would not let me either.

So the day before I flew into AMS (separate booking) and decided to try my luck at the Transfer desk robots. Surprisingly, that worked - somewhat. The first flight AMS-FCO was seemingly booked full and the first available seat was 24C. The second flight FCO-ATH was seemingly empty at that point with only a handful of seats taken. So at least I was able to pick a seat close to the front for the last segment, which was the important one.

Next day in AMS I checked in with KLM for the AZ flights, and requested a seat more to the front. Seems that my status had some effect as I was seated all the way in the front, middle seat blocked, with (what seemed like) a politician at the window seat. Must have had something to do with the EU stuff going on in Rome this weekend. Good point was though: Front seat, and the only free seat in the aircraft was the one next to me. I'd call that a success!

In FCO bus gates, turned out that the AMS-FCO and FCO-ATH flights were operated by the same aircraft and same crew. That was funny. We departed Rome with only a slight delay and landed in ATH 15 minutes ahead of schedule. I had planned to take a taxi out of the airport, but due to the early arrival I was actually able to save 40 euros and take the regular bus to the same destination. Doors open at 17:46 and with a sprint I was able to catch the 17:50 bus which was just departing but willing to stop to let me on.

All and all I was not disappointed in Alitalia, probably for the first time ever The seat selection was a mess but that was eventually sorted and I got great seats on both flights. Aircraft looked new and both flights were on time. Only thing I could perhaps be negative about is the fact that I received more food on the 45 minute LHR-AMS flight the day before (on KLM), than on the 2+ hour flight AMS-FCO on AZ. But I guess that's part of the game nowadays..

Thanks for the help everyone