I'm pretty good with electronics and pretty good at making poorly engineered things from China work a bit better.
Attached is a little draft schematic. I just have to solder a wire to either side of the on/off push button on the controller and connect it to my circuit which will "press" (connect) the button on a call for heat, and "press" (connect) the button again to turn it off when 1) the heat need is met and 2) at least 40 minutes have gone by. The 40 minutes is because the stove won't start again for 30 minutes after it is turned off. Since I'm not doing anything different, really, then if I were to stand next to the stove and turn it on and off manually, I feel it's a safe mod.
The way it works is when the thermostat calls for heat, the NO contacts on the thermostat close and the electrolytic capacitor is charged to pos 12V as current flows through the relays' windings. The pulse charge current through the the relay will close the relay switch momentarily (closing the push button connections and turning the stove on). (Once the capacitor is charged, it won't allow any current to flow). The charge current will also trigger a time delay relay that won't connect the circuit it is in, until 40 minutes have gone by.
When the temperature on the thermostat is met, the NC contacts close on the thermostat and connect the capacitor to neg. If 1) the NC contacts are closed and 2) time-delay relay is closed (40 minutes have passed), then the capacitor will discharge, again activating the relay switch momentarily, and the stove will turn off.