Don't know exactly how the Lilli switches to A/V input, but I just worked out the same thing using a Cheese Box. I used that because my LCD controller is VGA in only. The Cheese Box converts the A/V signal to VGA.
It was very simple because the button that switches from VGA to A/V input is the latching type. Meaning that when you press the button, it latches two contacts together, and displays the A/V input on the monitor. When you press the button again, it unlatches the contacts, and VGA signal is passed thru.
I took the switch off the Cheese Box. I connected two wires to where the switch used to be, and ran them to a relay. The relay is triggered by the reverse lights on the car, so when you are in reverse, the contacts are made at the Cheese Box.
I added a 3 position switch to allow for; A) Relay gets trigger from reverse lights; B) Relay gets trigger from constant 12V (so I can turn camera on without being in reverse); and C) No signal to relay (so I can be in reverse without the camera being on).
I suspect the Lilli changes from one input to the other by pressing and releasing a button. Then you'd have to press and release again to get back to the other input. Is this the way it works?