An Umbrella is a pretty big bird that requires a pretty large cage.
We have traveled with as many as 3 small parrots, each in it's own cage. Currently we have only the Grey and he has a small travel cage. We removed on of our recliners and use a wod TV stand to set his cage on and to stor his fod, cover and some extra toys.
As always you need to keep tham out of direct sun light and drafts. You have to be careful of fumes and vapors in a closed up space like an RV. be especially mindful of teflon pans getting to hot on the stove. Normal cooking in them is fine. I would not get one hot enough to fry though. Teflon when heated too much will give off a fume that is deadly to a bird.
The problem that I can see with a cockatoo is that they do tend to be screamers and they are pretty loud. If yours like to scream, it could be a bit disturbing to your fellow campers.
We have been camping with parrots now for almost 20 years.
Amateur Radio Operator (KE5DFR)|Full-Time! - 2012 6.7L Ford Crew Cab Dually -2013 HitchHiker Champagne 38RLRSB - Travel with one Standard Schnauzer and one small Timneh African Gray Parrot