Whether you consider your RV a home or a vehicle, gun transportation laws still apply. And these laws vary from state to state.
Here are two links that should help you understand the different rules and regulations:
NRA-ILA | Guide To The Interstate Transportation
TruckVault - Gun Transport Laws by State
Be espeically careful in more restrictive states. In NJ, breaking most gun laws can land you in jail, and many of those violations come with a 3 year manditory sentence.
As mentioned above, it wouldn't hurt to have a destination to go to (shooting range) in writing. Also doesn't hurt to call the local municipality and seek clarification (although they don't always know the gun laws well).
Also - if you have a CCW, that is a huge help, since then you are covered under the FOPA laws regarding gun transport. This gives you a bit more protection should you get into a situation where a local municipality does not understand their own state's gun laws.
Be careful.
-Z-man