That's a good looking stove. Obviously, you've diagnosed the problem -- there's just not enough draw.
If you could make an adjustable vent on the front door or on the bottom of the barrel, you could slide the vent open or closed as needed, that might do the trick.
One of the issues might be the large size of the burn chamber. Fresh air has a higher percentage of oxygen (O2), whereas spent air from the fire is full of
carbon dioxide (CO2). That extra carbon atom makes it heavier, sinking down and smothering combustion. Your fresh air may be rising above the layer of CO2 that sinks to the bottom of the chamber -- it just floats up over the top and goes right up the chimney. Some sort of baffle at the top of your burn chamber would force the air to mix a bit, or reconfiguring the chimney so it draws from the bottom of the back of your burn chamber rather than the top would draw air right though the burn zone rather than up over the top of the coals.
Short of that, perhaps lighting the fire right at the front of the burn chamber next to the air intake would help. The warm CO2 would rise and go out the chimney, and hopefully not smother the fire that is being fed by the fresh O2-rich air.
Best of luck.
"The rule of no realm is mine. But all worthy things that are in peril as the world now stands, these are my care. And for my part, I shall not wholly fail in my task if anything that passes through this night can still grow fairer or bear fruit and flower again in days to come. For I too am a steward. Did you not know?" Gandolf