We were decanted into Non-schengen departures -- if you are NS-->NS and you know your way around the airport, I think you could potentially make a very tight connection as the arrivals decant into departures.

I went directly to the Piazza di Spagna ITA Airways lounge as it was directly there, but apparently the lounge agents aren't able to handle any IRROPs -- I know I was trying my luck here as some airlines like BA, KL, etc. are able to handle ticketing issues & irrops at their lounges. Then, I went to one of the transfer desks to ask for alternatives -- this is where I got the extremely inflexible policy. To their credit, there are many ITA transfer desks -- I saw one in the Non-schengen departures, one before exit/transit security, one after entering the Schengen region, one in the main Schengen departures, all staffed with a number of agents so there wasn't a long wait anywhere. And yes, I did try to get my "ideal" rebooking on VY at each one, none successful.

I ended up taking the later AZ flight and moved my further flights even later as it was the cheapest option -- booking the separate VY flight would have cost 200€+ whereas changing my next flight was only 80€. Will submit an EC261 claim in a few days.

I would say 40 minutes is truly tight -- if you had all 40 minutes at the airport, I'd say it's possible, however, with the possibility of bus gates (quite high for Schengen flights I feel, 3/4 of my last AZ Schengen flights have been bus gates), the fact that AZ doesn't pad their flights too much, and frequent changes to the schedule (I had a LIN-FCO flight cancelled with about 2 days' notice, they rebooked me on the 5pm flight which meant I had to cut a meeting short). If you also need to exit Schengen and don't have an EU/UK/US/CA/etc passport, you're also at the mercy of immigration control lanes which can get somewhat dire at times.

The low frequencies, inflexibility, tentacle-like (and at times confusing) layout of FCO, and general poor handling of IRROPs (my checked luggage didn't make it to my later flight, even though gate agents had confirmed it had been loaded on my later flight), have put me off booking further ITA flights at the moment. Their prices are good, but you get what you pay for when things go wrong.