Is it possible to remove the Raduino and display from the case and connect to the USB port on your computer without the mainboard? The Raduino should powerup and the display should show the normal frequency screen.
If this does not work, then my next thought is the display or the Raduino board as the digi board supplier suggested. If it does work, then there may be a wiring issue in the case.
This is really a head-scratcher. I cannot see why the one Nano works and the 3 others do not. Are you sure that the Arduino programs are the same? When I read your statements it reads like you reprogrammed the original Nano with the files (sketch) that are the same as the 3 new Nano programs.
There is one other thing that you could try: reprogram one of the new Nanos with KD8CEC software that works with the 16x2 display. That would be in this archive:
The correct file would be in the uBITXV234 folder. The file for the parallel display is UBITX_CEC_V1.200_16P.hex
you will need the Xloader program to load the hex file.