If I were to want to take intercept / control of what the thermostat was currently doing I wouldn't touch the thermostat wiring at the thermostat.
I would use an AC input optocoupler back at the HVAC unit for each of the controls Fan, Heat & Compressor. Disconnecting them from the HVAC unit.
GPIO input the output of the opto's, perform what ever calculations needed using the Arduino and control the Fan, Heat & Compressor using relays. I would not use the cheap "for Arduino" relay boards on my HVAC system, I'd use something more reliable. But that is just me.
Note: There will be no need for the "C" terminal at the Thermostat.
That's my point. If you select a low current relay or P-MOS option, a screen with a low-power mode and a switch(or gesture sensor) on the backlight, and an Arduino on low power mode then there's no reason you can't stick a couple 18650s in there and call it a decade.
Or follow what @johnwasser suggested and use 2 Arduinos and repurpose your wiring; that's a no battery solution.