Yup I think he is long gone on his trip already.
Intuitively, I would think that while the motorhome is on the road it falls under the motor vehicle laws for firearms transport.
And while it is parked, then it falls under the campsite / motel room firearms laws.
And of course every state is different. In some states the car has the same possession laws as the home. In other states the car requires the firearm to be unloaded and trigger locked.
California is not really worth visiting, in my opinion. Their rules are as bad as NYS, Mass., or NJ.
Oregon does not honor most out of state CFP's but it is an open carry state -- same is true of New Mexico. So open carrying in a visible holster while driving your vehicle there is ok.
Wash State honors most out of state CFP's. As does Montana. As does Idaho. As do Nevada and Utah and Colorado. I am not sure what the current rules are for Clark County (Las Vegas) Nevada are though. These keep flipping back and forth. In Arizona anyone can carry openly or concealed without any permit.
That's the Wild Wild West. I generally avoid going any further East.
As I said in my post #4:
We can continue to list individual state-specific laws that change over time while missing specific issues, but the above statement remains. Also, firearms laws are rarely intuitive.
Here is an interesting issue. Oregon has only state preemption for permit holders. Those carrying openly in Oregon without a recognized permit will have problems in cities, like Portland. Portland Ordinance 14A.60.010 explicitly forbids carrying a loaded firearm without license.
The state of TN extended the castle doctrine to your car. You can have a loaded handgun, AR and shotgun anywhere in your car.