Just so I understand correctly, you want to blow or pull the colder air that are in the other rooms into the room where the stove/insert is located? I just assumed and I am new to this that a fan would be blowing the hot air out of that room into the colder rooms. Thanks!
You got it, push the cold air (denser air, easier to move at floor level) towards the heat source. It forces convection. The toilet paper trick is a good one, try it out.
And do not aim that fan upwards but leave it blow the air straight into the lower level of the stove room. This will force the warm air at the top (the warmest) of the room to move toward the rear of the house and then return as cooler air. It is the same principle as using a ceiling fan. The coolest air in the room is at the walls and cool air moves downward. So, in the center of the room in winter months you want to blow the air up and not down. This helps the natural air movement and will keep the room the warmest.