first think about inefficiency in a stove, then think if a fan can affect that.
A stove can be inefficient/wasting energy in 2 ways, #1 failing to burn clean and wasting energy as unburnt fuel(smoke), #2 wasting excess heat up the flue. unclean burn has various causes but that smoke going up the flue is wasted energy. one cause could be a cool stove, perhaps just lit or one dieing down, or just running very low and smoldering. Here a fan will make things worse by sucking more heat from the stove / firebox. So don't run a fan on a smoldering stove.
scenario 2, heat wasted up flue, occurs at the other end of the spectrum when the stove is running really hard. At this point a fan will help efficiency by sucking a bit more heat off the stove and into the room.
So yes a fan can help efficiency if you a running your stove hard, or it could hinder efficiency if your stove is just lit or running very low.
Unlike a wood furnace that blows across a flue gas heat exchanger in the flue gas path, the stove fans blow on the stove to cool the stove. The wood furnace situation where you can extract wasted flue gas heat has a real chance of increasing efficiency. It is the theory behind the magic heat device as well. Cool the flue gasses to the minimum required to increase the efficiency. Zero unburned fuel emissions and cold flue temps accomplish peak efficiency right?
The BK thermostatic stove will increase burn rate to maintain a set stove temperature so cooling the firebox will just increase burn rate. You just get more heat.