The C-wire would need to be connected to the appropriate terminal (usually also marked C) on the HVAC board where your other thermostat wire connections are found. Often if a battery only thermostat is ever used, this wire would be disconnected at the HVAC terminals.
Is the wire connected or disconnected from the HVAC? If you are unsure, please post a photo of the terminals and we would be happy to take a look! Make sure to turn off all power to the HVAC before connecting or disconnecting any wires.
That would beg the question why can’t an automatic adjustment from Alarm.com at a scheduled time not do the same thing?
This is not a change from Alarm.com in this case. The schedules are locally saved and run. We have noticed that schedules seem to have a tougher time with a thermostat learned in on battery so a good test would be to re-learn on 24VAC and test.
Apart from fixing the C wire issue, do you think I could pair the thermostat with an external C wire and then unplug it after pairing and use battery? Or would it just revert back to power save mode?
I’ve not tested this circumstance over a long period but if I recall from previous troubleshooting I believe the batteries will be used up very quickly in this event.