Take a look at the rocket oven design which recently finished a kickstarter that Paul set up (
https://permies.com/t/87936/rocket-ovens-kickstarter ), it is a 6" system and it's not too big for baking. I got to start it up for the pizza party and it started easily with zero smoke and you can adjust the temperature. That's part system size and part how much you
feed it- you can partially cover the feed with brick and burn one stick at a time, or leave it open and burn 8 sticks at once.
Batch box isn't at all superior/better than a J tube, it is different though as it is meeting a different need at a cost. You can (properly) load a batch box with more
wood and close the door and leave it be for longer than you can a J tube. But it will measurably burn more wood than the J tube for the same resulting heat according to Donkey, and usually will cost more to build (properly) due to the door construction. Of
course you can spend more money on any design based on the parts you use, but the batch box simply has a few more parts. I think Donkey said the batch box will burn about 20% more wood than the J tube as far as efficiency (meaning more wood for the same heat), but it might have been someone else in the room at the time.
Not only does the batch box need to be properly sized, but loading the wood also requires precision to avoid air flow issues. If you stack too tight against the port you will have problems, and if you light the tinder at the door end you will likely get smoke back. J tube designs can be built far more bulletproof (or Gilligan proof to borrow Paul's terminology) but can also smoke back if a person tries hard
enough to do everything wrong or it hasn't been cleaned lately.
If you are getting smoke out the exhaust then the burn chamber temperature is low, from reading about burn tunnel temps in
https://permies.com/t/93105/Cooking-inch-batch-rocket-stove that would be the likely issue. Without proper sizing/spacing, sealing of joints, and insulation of burn tunnel/riser the temp will drop and/or the draw will suffer. The temps you are reporting are less than half of what you should get if built properly.
When visiting Paul and Jocelyn's place I saw a couple batch box feeds and a few J tube feeds all in service and they all work great if used properly, as they were well designed and built. I've previously seen
rocket mass heaters that were in poorer shape at another site (needing cleaning or with less than ideal exhaust setups) which were more temperamental or smoked at times.
The podcasts on batch box heaters has some good wisdom regarding "improvements" as well, lots of people try to improve a design without understanding that the original design took years of testing to evolve to what it is. Batch box designs are still being tweaked to try and match J tube efficiency. Under perfect conditions (like Peter being the person starting/feeding his custom designs) batch box systems are probably right there though. But if all the corners are cut to make something from the few supplies on hand, there really isn't an
answer beyond "rebuild it properly". I hope that doesn't come across as mean, but I do think this is the issue.