I have seen and repaired 'flexible circuit boards' several years ago, some industrial counter devices used this idea. The board was similar in appearance to modern membrane keypads but a bit more tolerant of heat from soldering !
I think your biggest challenge would be developing the flexibe glass required for the LCD panel ?
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The original Veroboard or stripboard is produced using a mechanical only process. Etched version are also made by us at Vero Technologies including square pad and ground plane versions. Doing a flexi stripboard or square pad board is straightforward, we would just need to know some numbers to make it worthwhile, but they would be more costly than conventional versions.