I've never heard of not being able to start it/drive it, as long as you have the proper paperwork. Once you have the Paid In Full invoice, and you have the truck registered and insured, its YOUR truck. I don't see that its any different driving that off the base than driving your regular vehicle off the base.
Of course they would put up a stink if the truck needs any actual work done before moving it. Usually they have no problems with dead batteries, and many times even help you jump start the trucks.
If they seem impatient, or don't want you to do anything to the truck, just check the oil, jump start it, and get it just off the base, then give it a better look-over before going far.
Jim