UPDATE:
After 15+ hours on phone with both Chase and Alitalia, Chase picked up the tab and rebooked on AirFrance. Luckily for us- one segment of the original tix were business class (cheaper than economy). Chase rebooked as business. AirFrance was lie flat and a much better flight!
very happy for you! I was almost jealous that I "only" got them to make an exception (or magically disappear the tenth infant), but then again it took me only an hour on the phone, so I guess that makes up for the time in business class!