Hi, I have a 2200 although no longer use it. They are nice looking items even if they are not "real" B&O.
The display is lit by two tiny lamps which do eventually give up the ghost. A friend of mine with a bit of electronics know-how sourced some replacements from RS Components. They are 14v 80mA bi-pin filament lamps type T1.75. The RS code was 577-516 but that was some time ago so may have changed.
Getting to the lamps is not difficult. If I remember correctly there is a pin on the left of the fixed display panel (just beneath the lid when it is opened). Push that in and you can lift away the panel. You have to take the volume slider off first and if I remember there's a small screw underneath it.
Yo do need to be handy with a soldering iron to do this but if you are or know someone who is it is a not a difficult job.
They came in a pack of 10 so I have 8 left. I'll not need them so if you want to give it a go send me a PM with your address. Happy to post a couple of lamps to you.