How to "push buttons" on voltage ladder?
You have many options open to you, depending on a) how small and discrete you want it to be, and b) what your level of expertise is.

The simplest method is just to use some devices to short out the buttons thus emulating a button press.  Options include:


Small reed relays
NPN transistors
Opto-isolators


A more complex option is to replace the whole board with a digital potentiometer.  The voltage seen at KEY1 is determined by the ratio of R873 to a combination of R837, R844, R849 and R860. It's basically running as a voltage divider.  You can use a digital potentiometer to replace all those resistors and buttons and just set the required resistance to get the right voltage at that pin.  You would have to calculate the voltages each combination of resistors would equate to and ensure that the digital potentiometer (or combination of digital potentiometers - you can chain them together to increase resolution) can give an accurate enough voltage for it to be seen as the correct button press, and that very much depends on the device reading the buttons.

Simplest would be little reed relays - fairly discrete and simple to wire up.