I'm new to ToyotaNation and also interested in this same solution. I love my back-up camera (aka "Rear View Monitoring System") for aligning my hitch to the tongue of my utility trailer.
In addition to the benefits noted by the original poster, it would also be nice to keep the camera on when I'm making trailer hitch positioning adjustments while in Drive, not just Reverse.
But since my camera is connected to my "multi-info display", I think it would be more than just sending 12 volts to the camera, somehow the display's brain needs to think I'm in reverse somehow? I'm not sure what that hacking might do to the rest of the car's computer systems?
Reading this thread
Reverse \ Backup Camera Multifunction Display Wires doesn't give me any warm fuzzies!
This is beyond me but if anyone else in ToyotaNation with right smarts has come up with a viable solution, I'm open to trying it!