[robocat]

> sledgehammer approach

A hammerier solution would be to control the temperature seen by the thermostat (ignore the difficult RF protocol).

A heating element and a temperature reading could control the heat seen by the thermostat.

I'm pretty sure you wouldn't need any cooling (Peltier or whatever). Just a heater and ambient cooling! Set the thermostat to a high temperature, and run the heater to make the measured temperature hotter: when you don't want the heating to run.

That said, I think hacking the RF protocol is geekier and far awesomer.