I would not go with a battery operated T-Stat
Google "120 volt thermostats" or "in line thermostats"
I have two. One is a "Dongle" it is a plug/outlet assembly with a cord and on the other end is a thermostat. Rated 15 amps it runs one of my space heaters where the T-Stat (electronic) got Fried by a power surge.
The other one I have is a Wall mount.. NOTE the connections are made BEHIND THE WALL.. But it's mounted on the back of a fireplace like yours..
About 30 bucks each
Took me a try or two to get all the right wires in all the right places on the Mini=Fireplace.
