Setting in the radio
I installed the Pioneer AVH-600EX, but the Kenwood should be the same in terms of the rear view camera.
I leave mine on all the time to see my toad while driving.
You should not need to change any wires.
The wire that connects to the reverse lights is just used as a trigger for the radio system to automatically engage the rear view camera when in reverse. I did not need to change any wiring to get full time rearview camera. I simply hit the camera icon on the main screen on my Pioneer.
I would suggest reading the manual on the Kenwood.
If you are not getting the rear camera at all (including when the RV is in reverse) I would first check the settings on the radio. On my Pioneer, I always have to re-enable the camera in the settings after it has been in storage (i.e. meaning I disconnected the house batteries).
If you have it properly enabled in the radio settings and still do not get the rear, it is most likely the video switch box. They use a video switch box that takes all 3 camera inputs, and then outputs to the single cam connection on your radio.
Do you get a camera picture when using either of the turn signals?
If yes, try swapping the rear view camera and that turn signal camera at the switcher box. If you now get rear view camera with the turn signal active, you know that the rear input on the switcher box is bad. If you do not get the rear camera, you can now conclude that either the rear camera has an issue, or the wiring between the camera and the box.
Hope that helps a little.