Why not just use one of the battery powered LED stick up lights in the closet. Some LED fixtures come with motion detector. Turns on when you reach in there.. Batteries will almost last forever with LED lights and then you won't have to worry about safety of running extension cords around the wall.
ALso you probably have not looked at your intended 120VAC RV receptacle you want to tap into. The standard RV receptacles are the thin ones with daisy chain pinched wire installation type. No easy way to tap into one of those...
just a couple of my thoughts.
Roy Ken