Beautiful. Thanks.
You had to ask me all of that extra information to tell me the stove sucks?
Not really. . The stove isn't the most important part of the system. The most important part is the chimney. We don't know what that is. I'm going to guess it is not optimal.
If you have a large diameter, tall chimney that requires lots of heat, no stove will work efficiently. The rest of the system is what determines how much heat you need to let up, determining how long it will burn.
Your first question of operation depends on the chimney, heat output required, and many other factors. Atmospheric air pressure and outdoor temperature being important factors that change settings.
The flue damper isn't an insulated device that holds heat down in the stove. It is a variable resistance that controls the velocity of rising gasses. It is used to control an over drafting chimney. We don't know if you have an over drafting chimney. Weather conditions can require its use when cold and high pressure (more closed to slow a strong draft) and warm weather with lower pressure area may need to be wide open, not having enough draft. It is not a stove control. It is a chimney control that affects the stove by slowing rising gasses, which decreases draft, which slows the air coming in.
Try leaving the flue damper open more and closing the intake air vents to control the fire. You should have plenty of coals in the morning after an 8 hour burn. A burn is considered over when a match is needed to restart the fire. I've had stoves that size with coals in them until the next afternoon to restart the fire. Is there plenty of ash in the stove? Keep at least an inch on the bottom. You should be able to remove ash each morning from the front and rake the coal pile with charcoal and a little ash ahead to build the new fire on and it will take right off.
The reason you're watching exhaust temp is to keep the inner flue temperature above 250*f to the top.
Your surface thermometer reads about half the inner flue gas temp. So 300 is actually 600 internal temp, and you're guessing it cools to 250* before exiting at the top. This may be so with a tall 8 inch chimney allowing expansion and cooling of the gasses. An insulated liner the same 6 inch size of the stove allows you to keep the flue gas temp much lower since you have less cooling to the top. This allows you to slow the fire, radiate the heat inside instead of using it up the chimney to keep it clean and to make the stove go. This prolongs the burn duration and you will still have a fire in the morning. That's the first thing to do to make it more efficient. Size the chimney to the stove.
Don't worry about keeping the pipe temp high at all times. The 250* critical temperature to the top is to prevent water vapor from condensing in chimney when smoke is present. This prevents the flue walls from getting wet allowing smoke particles to stick, forming creosote. In the coaling stage there is no smoke, so temps can be lower. Only be concerned about flue temp while smoke is present.
Get a temperature reading on the elbow before entering chimney and let us know the diameter and height of chimney so we have an idea of flue temps. Also, is the chimney interior or exterior?