Thanks duncan, the knowledge that this was just lightbulbs was enough for me to hack a fix for my own unit. The result is not perfect but it's functional. If you have a small amount of soldering skill and a very basic grasp of electronics then this hack is very easy:
The problem is that there are two lightbulbs behind the LCD screen that illuminate a piece of translucent plastic. The bulbs are connected in series, so if one blows they both go out. My fix involves simply connecting a piece of wire across the blown bulb thus allowing power to reach the good bulb. In theory this alters the electrical properties of the circuit and may end up in death of the other bulb or worse damage. I've had about an hour's use out of my unit since doing this with no ill effects (it came with a car I just bought so I don't mind too much if it dies).
You only need the front panel. Unscrew the four screws on the back of the front panel and prise it apart. It's best to do this with the silver side down as there's a bunch of bits of plastic for the buttons that will fall out all over your bench if you hold it the other way.
To do the investigation I removed the LCD panel but this is not necessary for this fix. Look at the picture:
(On the pic, the LCD and bulbs are removed, so yours will differ a little from here).
The bulbs are wired at ba and bc. Connect your multitester across ba and check the resistance. If it's infinity, that's the blown bulb. If it's less (e.g. about 5 ohms) then that bulb's fine. Do the same across bc. Then, on whichever is blown, get a small piece of wire and solder it across. For a final check, connect the tester across A-C on the front panel connector (that's the top left and bottom right pins). If you see infinity, you either fixed the wrong bulb or didn't make a good connection.
Obviously, don't get solder on any other parts of the circuitboard or you're creating new circuits!