The last (first entered) photo is of some "Apple Boxes". The name for them was coined by the early photo and movie industry for stackable spacers to raise someone or some thing any height in 1" increments between 1" and 15", simply by stacking one or more of these to achieve the height desired. Each is 12" X 20" and a set contains 1", 2", 4", and 8" heights, so any height can be achieved, simply by stacking the right combination of these.
I built my two sets of these for my photo/video studio from a bunch of Baltic Birch drops that I had in my shop, so no money at all was spent for materials. The commercially available versions are made from construction grade plywood and pine and are very heavy, and splinter and break very easily. I wanted mine significantly lighter, but stronger, so used 1/2" thick BB for the top and bottom, and 3/8" for the sides, with 3 partitions inside to transfer the weight from top to bottom. The corners were box jointed, because I can do this easily using my Incra I-Box jig, Freud SBOX8 blade set, and my Delta Unisaw. Only Titebond II was used to glue them together, no metal at all.
I have had a 350 lb man on top of a 12" high stack of these, with not even a crunch as he climbed up. Baltic Birch plywood is incredibly strong when compared to our pine and fir construction grade plywood. I was tempted to use just 1/4" BB plywood for the sides, but didn't have enough to complete them, so I went with the 3/8" sides (would have been even lighter). Three coats of poly later they were put into almost every day use. They are about 1 year old now, and still in perfect condition.
I like to make boxes, and frequently make them from Baltic Birch. Almost all are made in a similar way, if they will be just tool boxes for specialty tools, camera gear, lighting equipment, etc. I have never had one break, even when dropped on one corner. Scuffed maybe, but not broken.
I've also included a few photos of other boxes that I've built using the same basic process. I doubt that going any heavier in Baltic Birch will be necessary for your plan, especially if you add any partitions to your drawers.
Charley