This is sooooooo confusing,lol. Have you tried to run two wires from the reverse light wiring (just hook them up to the blue clips), and just attach those to the bulb?? Another possibility is that the voltage is there, but the current is really low. In fact so low that it wont light up the bulb.
p.s. you must be doing a lot of running around constantly putting the car into reverse, running to the back of the car to check. ALSO, maybe you have already managed to fix the
problem at some point, but you didn't realise because you forgot to switch on the ignition.
Mig