OK, nevermind. It's not a straight through connector, so I won't be able to fix this. Thanks anyway everybody!
JLCPCB (and others!) can manufacture two-sided flex PCBs like this, and they're not too expensive. Right now, 100mm×100mm (3.94"×3.94") cost about 14€ + shipping for the minimum five, but the price does get up for anything larger. It isn't difficult to reproduce the flex PCB in say EasyEDA, either, if you just find out exactly how the pins are connected.
I recommend you start by taking a picture of the existing flex PCB on top of graphing paper (or similar, with a rectangular grid), making sure the camera lens is perpendicular to the paper and centered, to minimise distortions. The size is the determining factor for the flex PCB size. Then, check the connectivity using a multimeter in diode or ohms mode. Make a simple schematic to document them, then convert the schematic to a two-sided board, and draw the traces and vias.