It's an awesome hack!

It seems like the easier hack would be to put a peltier heater/cooler under the thermostat then control that remotely to assume control over what temperature the thermostat sees.

The link to the exact model of thermostat isn't working, so I don't know how amenable its design is to this approach, but the thermostats I've used are generally wall-mounted and putting a heat/cool source under them wouldn't be too hard. You'd need to make sure that you didn't send both the heat and cool into the thermostat, but that's a simple positioning problem.