wow, sounds like an amazing trip!!!
your destinations have a bit to do with your desired setup, but less than you think. the only problem i see is USA voltage is 110, European Voltage is 220. so i dont know how your shore line will work, you'll probably need 2 one for usa camping and one for european camping.
to figure your capacity, you need to list the electrical items that you will use, the time you intend to use it, and how much electric it consumes for the time.
then you need to figure out how often you will be able to charge you battery bank.
then add up some losses for inefficiency and you will be close to your battery bank size.
what i've seen on the forum is that 24v 400ah is about as big as these guys get.
the trend is for lithium iron phosphate batteries, solar, and an inverter.
you are going to have to decide on appliances, voltages, because of your trip. an american blender will not last long plugged into the EU grid.