I hope you father gets better soon. The pictures are very helpful. You have good welding skills! I can see why you chose metal for the prototype. As mentioned above, an insulated burn tunnel and the lower part of the heat riser will degrade metal quickly in a "permanent" installation. But back to our discussion of getting your prototype rocketing!
I understand why your feed tube has the opening increase to 13". It's the shape of the preformed metal that you are re-using. If this larger 13" diameter starts to close to the combustion zone, the slower air movement in the large diameter area will make it easier for the fire to creep up the sticks/fuel. Typically the feed opening will be keep the same CSA as the burn tunnel along it's length. This will keep the air moving and cooling the fuel so the fire stays put in the bottom.
The first picture shows a small hole at the bottom in the smaller diameter area of the feed. I believe you had described this as an air intake. This may not be having the desired affect for your prototype. By having this opening it greatly lessens the flow of air coming down the feed. In fact, it makes it easier for the feed opening to act as a chimney leading to the smoke-back you have seen. Combining this with a tight fitting lid over the end of the feed opening has the potential of actually being dangerous. Let me explain the possibilities. With a tight fitting lid, as the fuel is consumed unburnt gasses and heat start collecting in the feed opening. If the unburnt fuel gets hot enough it could start outgassing. Combine that with enough heat and it only needs a breath of fresh air to ignite, like taking the lid off of the feed opening.
All is not lost. A simple two stage test can check this theory. First, block off the additional air intake at the bottom of the feed so the air will be pulled in past the fuel. You could try a small fire to test just this change. Second, if you have some additional duct/piping in the 5" diameter range, same CSA as burn tunnel, stick it down into the small diameter section of the feed opening. Also limit the length of this test feed tube to 1/2 - 3/4 of the height of the current feed opening.
These steps should change its behavior dramatically, even with the 5 foot test chimney. Try this along with some of the other users suggestions and see if the rocket comes alive for you. I look forward to hearing your results!