Hi there,
this is a very simple system.
The lights can be undone from the outside with a cross head screwdriver, to check the bulbs. The next suspect would be the switch. It lives on the right side of the gearbox (underside), with a cable bolted to the gearbox in 4 places. Number 3 here:

- L300_rueckfahrschalter.jpg (58.31 KiB) Viewed 10857 times
Follow the cable to the 2 connectors on top of the gearbox.
Undo the connectors and short the two wires on the car side of things. That should make the reversing lights come on (with ignition on) if fuses and bulbs are OK.
On my van there was an additional issue as the two big white connectors living in the battery compartment were having bad contact as well. No damage done if you unplug them and put some contact spray there before replugging. You find the plugs when you follow the cables that live near the Diesel filler hose (good to see from the underside of the van).
The switch has a M12X1,25 mm thread size, thread is 11 mm long, with the pushpin (diameter 4.5 mm) protruding another 8 mm.
Spanner size is 19 mm, it is mounted with a 1 mm seal.
The original Mitsubishi number would be MD730711.
Replacement switches are a bit dear in Europe, and there are no aftermarket copies available. I was happy enough to have a spare from a wreck.
From the looks of it there are very similar switches around, just with a single 2 pole connector instead of two single pole connectors.
I measured them up, but as you loose some gearbox oil when changing the switch I did not test it by now.
The older MD730709 may fit as well, cable/plugs to be adapted.
The best offerI found was a replacement by Facet:
Facet 7.6047
They say it is identical to:
Hyundai 93860-31100 , 93860-14500 ,
Mitsubishi MD-730709 , MD-700198 , MD-703797
Andy