The cheap LCD will probably not be able to produce enough light to brighten up the main display, and if the two panels are not of equal size, you will need to add some depth to the unit and come up with a projection system to "blow up" the light map to cover the entire image.
How easy would be to place a cheap LCD at the back of the main screen and mirror the same output (in horizontally inverted mode)?
Technically it might require synchronizing the frame latency differences between the two devices, but would such a hack improve the perceived quality?