On rigid displays such as lcds you come across designs using a type of rubber strip that is conductive only from top to bottom, not left to right (think a stack of slices of conductive and insulate material). This rubber is placed between a terminal strip on the screen, and similar pads on the pcb, and light pressure is applied. Where the slices make contact with a conductor on the display, and a pad, current can flow, but the adjacent pins are still electrically isolated from each other.
I have no idea what the material is called I'm afraid, only came across the construction method stripping down some scales recently.
Maybe something similar is possible with this display? It appears there are two holes in the FFC for attachment / clamping.