Remove the back label and the screws

Remove the internal screws

Cut the trace that goes from the bottom part of the AG13 coin cell (bottom left in the picture)

Carve a hole in the case for the button

This is how the button looks from the inside

Cut the corner of the pcb and solder the button wires

Close everything (my caliper only had 4 screws, so the screw hole near the button does not have a screw)

Final result