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 https://www.ebay.com/sch/i.html?_from=R40&_trksid=p2054436.m570.l1313.TR0.TRC0.H0.Xself+lock+5.8.TRS0&_nkw=self+lock+5.8&_sacat=0
This is how the button looks from the inside


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

Final result
Nice idea! I made a modification for my calipers which have a thumbwheel https://hackaday.io/project/178318-digital-caliper-battery-hack Might be of use to someone.