You can try the conductive epoxy on the market. Used to repair rear window defrosters on cars so I would head to an auto parts store.
Someone experienced in soldering should be able to tin the wires on your new switch and then tack the wires down with little (no) risk of damaging the PCB.
also, any attempt to add clips or springs will end in sadness because it is so difficult to secure something like that. A pull on the wire will dislodge it so you'll have more work than simply disassembling it.
Be careful with the conductive epoxy, it can make a mess if you are not used to transferring it so I would practice with some wire and a penny to make a neat bond before moving on to your device.
also, how old is that device? We were just chatting on another thread about how old the Microchip 16F628A chip is and now I see one in the wild. Also, how are you reprogramming it. Loading new code into the microcontroller or just pushing a button to let the chip adjust to new settings?