I built exactly what you are describing several years ago. I used a 4'x8' sheet of 3/4" Celotex (rigid orange colored foam with shiny foil on one side and dull foil on the other side). 1 sheet will make a 2' x 2' x 3' tall smoker (including top and bottom) with zero waste.
The build couldn't get more simple, all you need is a straight edge and a razor blade. I cut all of my pieces with a 45º bevel on the edges. Then I laid them all side by side (bevels and shiny sides down) and taped them all together with foil HVAC tape. Then it was just a matter of folding the box together and taping in the top and bottom. The beveled edges allowed everything to lock together very well and kept things square. For good measure I taped all of the seams on the inside as well.
I cut a large door in one of the side panels, again with a 45º bevel around the parameter. This allowed the door to seal well when closed. I taped all of the exposed edges of foam around the door and the opening. A single piece of tape along one edge served as the door hinge and I used one or two small pieces of tape to hold the door closed each time I used it.
For a vent I used a 3" chimney cap that I fit a damper into. I just cut a hole and taped it in. It worked great but in hindsight I think a drop in floor register would have been easier and just as effective. Thermo probes were simply poked through the sides.
My intended use for this smoker was for cold smoking sausages. The sausages were hung from metal rods that were merely poked through one side of the smoker and out the other near the top. I dug a fire pit in my back yard and buried 6' of 4" pipe. I cut a hole in the bottom of the smoker and simply placed it over the pipe where it exited the ground. It worked fantastic.
I also used it for hot smoking as well. Over the years I fired it with a small charcoal pan, an electric hot plate and a small propane stove. All methods worked quite well and it held temp like a dream, even in the winter. I smoked hundreds of pounds of deer sausage, dozens of trout, a few wild turkeys and a handful of ribs and butts. Fish and sausage were hung from the rods while other items were placed on a free standing wire rack that I had that fit perfectly inside.
It lasted about 6 years outdoors, unprotected and survived many storms including a big ice storm and a few tree limbs falling on it. The best part of it was whenever it got damaged it was super easy to repair with HVAC tape. I've done other builds since then including a freezer build and an awesome reverse flow, but that little Celotex smoker is still one of my favorites. One of these days I may have to build another one and run it with an AMNS.
I think I am in agreement that the side box is probably not needed. If the AMNS does get it warmer than you want, you can always put a cake pan full of ice and rock salt inside.