OK, a couple things I'm tossing around here...with your smoke-box output under and level with the burner/flame, I suspect it is breaking-down the smoke before it can get above the heat source. Your previous mention that when you open the lid and you can see smoke starting again, this may be caused from it getting some fresh, cooler air near the smoke-box outlet to the smoke chamber...I'm just not sure on this.
Something I missed earlier, about no air into the smoke-box...if this is the case, and you have a fairly tight-fitting door, you'll need some air holes on the lower portion of that cylinder (either on the bottom, or on the side) so your smoke wood can smolder once it gets going and you close it all up. A good system will adjustable intakes for the smoke wood. This could be a cross over a hole with a stud or bolt welded to the center and a disc with a hole for the stud, then a nut welded to the disc, on center. That simple vent control can be turned towards or away from the wall to increase/decrease air flow. With the cylinder, you could add a 2" piece of pipe to mount the intake air control onto, so it has a flat surface to contact.
If having a controllable air intake for the smoke box doesn't allow a strong smoke to continue when the lid is closed and the flame is on, then you need to release the smoke well above the heat source, as it could be breaking down due the burner's heat.
I see that you have possibly a small propane burner for a smoke-wood heat source, as there's a gas line branching down from the cook chamber fuel line, so it shouldn't be an issue keeping the smoke wood hot enough to smoke, but without additional air for the wood, it will be a very light, almost unnoticeable smoke, unless something changes which increases the air flow through the smoke box...opening the lid may be causing just that.
All that said, if you don't have fresh air into the smoke box, you'll need to work that out before making any other changes. Turns out, this should be the simplest fix as well. Then proceed with additional mods if this doesn't get a good smoke rolling through the smoke/cooking chamber. The beauty of having adjustable air flow to the smoke box is that you will have that to work with adjustable heat for superior control of the smoke output. If the smoke is breaking down from cook chamber burner heat, none of it will matter until that issue is corrected, if in fact that problem exists.
I think I've got my head in the game now...I was wickedly tired when I posted last night, and I don't think I read everything thoroughly...my bad.
Oh, one last thought after re-reading from the first post, you may be able omit additional mods inside the cook chamber by closing off holes for the gas/flame where it's closest to the smoke box discharge (self-tapping screws, if they're small enough, could be easily removed later)...this would reduce the heat near the smoke and hopefully prevent the break-down of smoke from high heat...easy fix there, but it creates a dead-spot (no heat) on your cooking grate of, say 6" x 8", give or take. When you're smoking, you may want to close the right-hand cook chamber vent and just vent with the left side, to carry more smoke throughout the cook chamber...but you probably already thought of that.