PaulRB:
Next idea: this display placed at a 45 degree angle, illuminated by a bright LED, perhaps a colour-cycling RGB, with a focussing lens placed above. What do you think?
I don't think reflection will work very well but you never know.
I pulled apart one of the little Nokia LCD's and as suspected it does use the horrible rubberised connection to the PCB but I think it would be possible to cut out the centre of the PCB under the display matrix and shine a light through (I did not remove the reflective backing) the LCD mounted in a clear plastic/glass package.
Another idea would be a LCD like this (or this) that uses a flat ribbon connector so may lift off the backing board and work with light shone through it.