Hi Knold,
you should be fine with a bag slightly smaller then the usual Pilot Bags.
Flying on the 604/605 myself, I can tell you that there is no forward lav and the rest and all of the storrage possibilities back of the Cockpit will be used by the flight attendants.
Depending on your size and the way you like to have your backrest you should be fine to store it behind your seat or in the aisle next to your seat.
I personally have the Alpha 2 from Tumi which is perfect and suits all my stuff I need.
Have fun