You may be able to have a good electronics shop pull the components out of one unit and place them in the other. The backlight works on 50 volts, and there is a small transformer in the unit. If the transformer is out, then a good radio shop might be able to replace it. Any company that does board work could do that.
If the back light is out, then we would try to either;
1> pull one out of the old unit,
2> try to get a similar flat one and black out the areas that you do not want light coming through,
3> replace the transformer with another one, these should be obtainable.
If you have a good radio solder tech., then you should be able to get this handled locally with no real trouble. Though a good small solder tech is harder to find today, because most just dispose on the printed circuit boards and replace them. Though some one in your area is able to do a solder job and tech that board.
They just need to hook 12 volts to it, trace the line with a voltmeter to determine the incoming voltage to the components and then in and out of that transformer.
Check your local electronics store, which most have gone out of business, or Radio Shack.
BCB