I had the same issue..... While there must be other ways of accomplishing this in the end I went this route:
www.ebay.com/itm/320842385347 Order warning: Takes 30days for delivery.
I tried using some capacitors but they didn't keep the camera
"live" long enough. I still haven't installed this timer yet but it
should work just fine.
The timer is activated from the backup light circuit using a separate
"hot" to supply the camera relay (part of the timer) and keep the
timer working during the time needed. Camera turns on immediately and stays on
for the time selected.
I plan on using the "hot", when the ignition is on, that supplies
the rear cigar outlet but have been to lazy to pull a wire from there to the
lift gate area.
On a side note I odered a second one to keep the stereo on when going from:
Parked with ign off to starting the engine. I just hate it when the AVIC 940BT
has to recycle for a momentary power loss.... This 10amp relay/timer can handle
that just fine.