Odds are it's the switch.

It's on the side of the gearbox (I assume you're UK as you've got a
316), IIRC it's on the passenger side of the box. Look for a big 26mm+
nut with a connector... That'll be the one.

It's quite easy to change, but you'll have to get under the car.

If you want to check it, put the car into reverse, then get
underneath. Take the plug off the back of the switch and you should
find 2 connectors, put a multimeter across them. You should see a dead
short when it's in reverse.

You don't need the engine running, or ignition on for this, in fact
put the keys in your pocket for safety!

Hope this helps

Dodgy.

My '94 E36 has this same trouble. I get 12v to the lights when the lights
are not installed, but the voltage goes away if the lamps are installed.

I think the trouble here is that the switch is dirty, very dirty. Voltage in
a circuit is dependent upon the volts applied, the resistance, and the load.
If the resistance is high or the load is great, the voltage will drop. I
think this is what is happening, the load and resistance are conspiring to
drop the voltage. Specifically, the resistance in the dirty switch contacts
is the trouble spot. The load is high by design -- lights are load -- but
the load and voltage are constant (the lights and the voltage have not
changed) so the remaining variable is the resistance. I've noticed that my
back up lights work on occasion (I noticed them on one day when my wife was
backing out of the driveway), so there must be a variable that includes
environmental conditions -- the resistance is acceptable in some conditions
so the lights come on.

I am not aware of any adjustments to the switch, and I have not been
motivated to find the switch itself. I suspect that when the switch is
removed, you may find a way to shoot it with contact cleaner and revive it.
The switch will be located on the transmission.