This has been discussed here on the forum many times and the most common and accepted answer for a shotgun in the RV seems to be that; it can't be within reach of the driver, it can't be "sawed off" (or have a barrel of less than "x" inches), and it can't be loaded with a round in the chamber.
As for your RV being your "home" and exempt from illegal search and seizure, the courts have ruled that an RV on the road (or movable) is not protected and can be searched without a warrant by a LEO if he/she has probable cause. If the RV is on blocks and can't be moved, then the LEO has to get a warrant.