A chisel of any type requires that you "force" it into the material, in this case "wood". Wood has directional grain which is present when it grew on the tree it came from ... unless it's a man made material like MDF. When you have enough force needed to "shear" away the material/wood, you also risk the material remaining will also crush, shear or split along the grain lines. It's too thin, too fragile. It won't work. :frown2:
A rotary cutter, like a router bit, requires only enough force/spinning power to remove small amounts of material each time the cutter rotates. This is less risky to the surrounding material. A CNC machine could be programed to make the "square holes". It may be expensive and beyond you budget, I donno?
A 1" mortising chisel will also cost a considerable amount and will be difficult to "force" into the material/wood without a very large press, or hollow chisel mortising machine. This operation you are describing is "material removal" much like drilling a hole. The problem is you can't drill a square hole without a very specialized machine.
The tool you show in the bottom photo is a "broach" used in metal machining. It will not work in wood because of the force needed and the fragile material surrounding it. It will break the thin surrounding material.
If you want to try this ... :frown2: make an accurate drawing having 4 of the squares on some wood, maybe Pine or other soft wood. Then accurately cut and chisel right up to the lines and see how well it works. The reason you asked on this forum was to get "good" advice from woodworkers with experience. If you chose to ignore it, that's your choice.