But looking closer appears to be an expensive dual power button (on/off) assy.
Your measurements would help.
View attachment 146626
A mechanical finger needs to be a firm short push with little or no gap and not a fast punch that might crack the colored PVC.
The solenoid and relay need a reverse diode across all DC coils to catch the flyback arc.
The force drops quickly with extension, so add a suitable extension for a very short stroke and about 10 to 20 watts for 1/4 second. (much longer and it gets hot) Then it should work for a 3 oz button. Adjust gap to minimum.