Backup Camera

Always tricky to figure out how somebody else wired it in. I would bet the camera power wire is connected to a backup light. And the easiest fix would be to run that wire down to the hitch wiring power wire. A nicer fix would have a pair of wires running from the back to the driver seat area so power can be switched on/off from there. Worth checking the cable from the camera to the display - it likely has a wire supplying the monitor power. Likely you could just disconnect the camera from the backup light and instead supply power to the camera-monitor power wire from some source at the under dash fuse box via a switch.

My back-up camera video screen is mounted on the rear view mirror and gets it's 12 volt power from under the dash. It stays on when the ignition is on, and is fed from the power wire that was originally meant to go to the dashboard stereo. This power wire gets it's power from the chassis. The dashboard stereo in mine is fed from the house batteries.

Mine is Auto On reverse or just push a button and on full time.
All my back up cameras worked that way. Would think this one should also.
We will know more if OP tells us make of camera.

Can probably pick up power from a vent fan, overhead or reading light, whatever's closest to the back wall where the camera is.
If you'll call or email Jayco they can send a wiring diagram, might help tell where you could tap in.

Camera could be wired to the reverse lights. Quick fix if that's the case is to grab 12 Volts from the running lights instead and just turn on the running lights when you want to have the rear camera on. Otherwise you will have to run 12 volts all the way back there with a switch so you can turn the camera on and off from the cab.

Mine is wired to the ignition and is on whenever the ignition is on.

Must be some control buttons. To make camera on when you want it and adjust brightness and contrast.

What brand and model back up cam?

The power source for the camera is currently from the reverse of the transmission. It needs to be sourced from the ignition. Pretty simple or any automotive shop or you if you are inclined, can do it, in my opinion.