Note that 'Muscle wire' needs to be heated to >100'C to shape change. You may need to think about thermal protection if the "mechanical button" has a low melting point actuator. It looks as if the heating current needs to be around 1.5A through the 0.012" wire (0.31 Ohms/inch), or 320mA (datasheet value) through the 0.005" wire (1.9 Ohms/inch).
Interesting looking stuff if you can get the mechanics, terminations (datasheet mention of diameter cycling), and drive circuit right.