We have done a number of trips recently with Master EfficientTraveller who is now 4 months but we took him to Cyprus at about 8 weeks.
I would definitely recommend checking everything in at check in and just using the carrier in the airport - it makes security a breeze as you don't even need to take baby out of carrier and leaves hands free for everything else. A blanket on a sofa in the lounge makes an easy place to put baby down.
We have a MaxiCosi Pebble too and have also bought a generic iSafe bag from Amazon. The bag doesn't provide any padding so we tend to wrap the seats in a few layers of bubble wrap. Given the seat is needed for the journey to LHR, that means a bit of wrapping in the drop off zone but it is achievable. We also normally throw nappies etc in.
We have a Babyjogger City Mini with carrycot which is only a bit more compact than the Bugaboo. I would definitely recommend a bag for the base wheels and we managed to fit the carrycot in a (very large) wheeled duffle bag with all baby's other stuff inside. While the carrycot is bulky, we found having a familiar place for baby to sleep very valuable and took baby to dinner asleep in pram giving Mummy and Daddy some nice relaxing meals. The Italians were very relaxed about babies in restaurants!