How can I use a transistor to "press" a button?
A glance at the TIP22 datasheet shows that this is a Darlington pair transistor, which means that it has very high DC gain (good), but at the cost of a very high Vbe_ON (bad).  The datasheet shows that the Vbe_ON is 2.5V, which means that the voltage across the 1K is only 2.5V max, which limits the base current to around 2mA.  This is probably not enough for good saturation.  In addition, the Vce_SAT for this device is about 2.0V, which may or may not be enough to look like a good GND to the button circuit.

I think a MOSFET like the IRF540N, connected as shown here (substitute button circuit for motor) might be a better choice
YMMV