If you can't run a second cable or have a monitor with a second input, it should be possible to fit a little gizmo to two cameras (one Reversing, one Rear view) that shows the Rear View normally, and then if it receives a signal (from the reversing lights) it will switch the video output to the Reversing Camera.
I think there is also a system that has a single camera that is motorised and on detecting a reverse light signal will rotate the camera downwards to act as a reversing camera (a bit like Door Mirrors that automatically drop when reverse gear is engaged).
either one of those systems would save running a new cable all the way from back to front (you would need to run a cable from reversing light to camera box though)