Thought this was in interesting question - kind of pointless and unnecessary - but I like stuff like this so I looked at a few wiring diagrams. It is actually looks possible to do this or at least on paper it does. The green/brown wire at the mirror connector is the 12-volt signal voltage from the reverse lamp circuit which triggers the camera to turn on.
You could easily wire one of your upfitter switches to apply 12-volts to that circuit but this will also cause the reverse lamps to illuminate and trigger the parking aid module if your truck is equipped with PAM. Additionally this would also power up the trailer tow reverse lamps circuit. The good news is that this circuit is isolated from the powertrain controls via the reverse lamp relay.
If I were to try this I would splice the output circuit from the upfitter switch to the green/brown control circuit for the camera close to the mirror. To prevent voltage feedback to the other modules and lights on the circuit I would install a diode in the circuit just before the splice. This will prevent the switch voltage from back feeding the circuit but still allow voltage to reach the mirror when the transmission is shifted into reverse. In theory, this should work... don't blame me if you break your truck, burn something up or otherwise render anything useless.
That includes driving your truck into something or someone while you are playing with your reverse camera while driving. I know, to most people it's no different then using a cell phone, a navigation system, adjusting the radio or putting on make-up, all while driving and these things are done countless times every day. That doesn't necessarily mean we should be doing them.
