Oh my got! This is most crazy plan I ever heard of. Why don't you just use usb camera, connect it to the computer, make very simple interface to sense reverse gear, and pass it to the computer (let's say through parallel port), then write quick app to dispaly the view.... I'm kidding, it's rather job for software developper. But it would be right way to handle it.
Anyway, you'll probably need rising/falling edge detector that could drive your screen VGA/AV button. Try to find flip-flop of JK type circuit, or some kind of logic state change detector. These have typically 5V or 3.3V logic signals on outputs, and I believe your monitor button basically switches the function upon shorting AV/VGA signal to ground. So this should work just fine.