What KBD are you Hacking apart, looks like ADB II? Got a pic of the controller and its connector?
Folding the sandwich underneath has always seemed a bad idea to me. I've only considered:
__ customized PCB made for the bottom layer
__ controller bodged to extension on PCB
__ lop off bottom layer and spacer
__ wrap only the top layer underneath at rounded off edge of PCB
That approach limits the number of cut matrix lines to the top, flex layer.
Older controllers were typically single sided, DIP component PCBs with wire jumpers topside. They can be easily attached to new PCB matrix at the connectors a/o by clipping the wire jumpers topside with longer wires soldered to bottom layer of PCB extension for support a/o connection bodgery if convenient.
Dunno, sleepy but this gets the notion across?
edit: keycaps top surfaces for curved rubberdome switch matrix config are likely going to be somewhat mismatched for a planar config?