Personally I would not mess with trying to repair this system. Instead I would check it out, then buy a new wireless rear-view camera system.
I have an aftermarket Voyager wireless BU camera system
in our MH. Instead of wiring the camera power to a backup light, I wired mine to be powered when the ignition key is on. This way I may monitor what I'm towing, the hitch, and still see when I am clear to change lanes.
To do this you have 2 options. Wire the power to the camera to a clearance light, then drive with clearance lights on.
I used the old camera coax to run power from the ignition switch to the camera, I wired 12V positive to the center wire, then used the braided shield wire for ground.
This solved the problems of locating a key-on hot wire in the rear of the MH, and how to run wires if a hot wire was not found.
If you prefer a wired BU camera system, it will be about half the price of a wireless system-just below the wireless system in that Amazon link.