Adding a second thermostat may be the way to go.
The existing thermostat is in the living room based on the assumption you spend a lot of your time in and around the living room, and so that’s where you want to be able to adjust the temperature to be comfortable.
If this is your situation, then moving the thermostat may not be the best solution. When you’re not using the fireplace, it could make it more difficult to get a comfortable temp in the main living area.
Instead, think about adding a second thermostat in parallel with the first. Put a thermostat upstairs (preferably a clock type stat, which can turn itself up & down based on the time of day, and day of the week), and set it at the lowest temp you want to maintain, say 60 degrees. And if you do use a clock thermostat, you can set it so that it only wakes up on weekends, and in the hour before you go to bed. This can save a few bucks in heating.
Since the two stats are wired parallel, either one can turn on the heat. If the downstairs stat is set at 70, but the room is 75 because of the fireplace, the upstairs stat will still turn on the heat if the temperature there drops below its setting.
On days when the fireplace is in operation, the upstairs stat will act like a low temp limit switch, keeping the furnace on just enough to keep the upstairs temp at the minimum level you set. You may even want to close the downstairs vents, since you may not need any additional heat beyond the fireplace.
On days when the fireplace is not in use, the higher setting on the downstairs stat will keep the furnace running until the downstairs is up to temp, presumably higher than the upstairs stat is set for.
Note again that since the two thermostats are in parallel, either one can turn on the heat. This means that both must be off (or set below current room air temp) before the furnace will turn off. So be careful about they are set, or you might not realize why it is warmer than you think it should be, and the furnace is still running.
But if you keep the second stat set low, this arrangement can work quite well. The second stat is fairly cheap (much cheaper than trying to zone a house that wasn’t plumbed or ducted for it), and easy to install.
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