On your close-up photo of the switch - you don't want to solder to the switch - you need to connect to the solder pads.
Not the gold switch membrane or the silver bits within the switch itself.... but those small silver (solder) corners that are visible (L-shapes) just outside the four corners of the 'gold' square.
From the image as it is shown above, it appears the pads you need to connect - are the top and bottom L's on the RH side. of the switch body. The tracks may run under the switch - but they look like the ones you want)
The switch itself is irrelevant - you are bypassing the switch with your wire and (relay) or whatever.
(I just can't press UNNOTIFY!)