Can you access the pads on the PCB that the button makes contact with when it is pressed? Solder a wire to those two pads, and then have a relay make that connection. Will require a little soldering.
Or, build up a servo with an arm that can press the button for you.
Or, spend $20 and get a programmable thermostat
Or, check with your heating supplier, see if they offer home energy assessments and provide free programmable thermostats.
Massachusetts offers free home assessments, in the past they came in, sealed around pipes going into the attic, did some other stuff, gave us free power strips and LED lightbulbs, seals to go under faceplates of wall outlets, etc.