How about adding side illumination to the edges of the display instead. Not as even, but will work well enough for most applications, using only 2 thin SMD led chips per side. You can get the chips new, as modules you place next to the display, or grab them off an old flat screen LED backlit display easy enough, though often those are a multichip module per pack, often having around 9V forward drop, but really bright at 20mA. done that a few times to a LCD display to make it night readable, not even, but better than nothing, and pretty much the common method for these displays.