A.J.
The flight to Greece is grueling. I would get baby a seat you will all be happier.
I have flown internationally with my children. Use your carseat, change diapers DISCRETELY in your seat - not on the tray table. If it is a messy diaper I would definitely take baby to the bathroom. There is just enough room on the sink area. Request special baby meals. Check your stroller at the gate.
Make sure you have a carseat that will fit in small european cars. Do not trust the european ones. They do not always have a 5 point harness. Some areas of Greece have the highest auto accident fatalities rates in all of Europe so a good carseat in imperative.
Regarding the time change, be prepared to be awake in the middle of the night to feed your child. This will probably only be the first couple nights. Your baby will adjust, but not completely, until you are just about ready to leave!
Have fun. Greece is an incredible place to visit and make sure to eat figs and baklava while you are there!
A.