So you want to, if possible, use just one display, and have it automatically display what the reverse camera sees whenever you shift to reverse?
Does the display have more than one input on it? And of what types are they? If it doesn't, you could have some switch that switches the input from the computer to the reverses camera, actuated by the reverse lights or some other circuit that closes when you shift to reverse. Then when you shift out, it would return to the computer.