I would probably just go about this by heating/cooling the thermostat itself rather than messing around with radio signals. Put a little box around it and something that could control the temp in the box, like a little peltier element. When you want the heat to run, cool the inside of the box. When you want it to stop, warm it up. Etc.

But then I build thermal control devices for fun so maybe it just seems like a much easier method to me.