How will this not work? Let me count the ways.
1 - Your boxes are apparently made of "thin" aluminum, which presumably means ~1 mm thick. These walls will flex as the box is moved, and you cannot guarantee that the two sides of the Peltier devices will remain in contact with the aluminum.
2 - Assembly would be a nightmare, as you would have to mount the TECs on one box, then slide the two together. And since there is no physical contact between the boxes, there is no way to hold the TECs in place.
3 - If thermal contact is made, the thermal conductivity of aluminum is inadequate to spread much heat laterally away from the TECs, given the very small cross-sectional area which the small thickness implies. This means that the immediate areas on each side of the TECs will get very hot and very cold.
4 - These hot and cold areas (roughly 40 mm x 40 mm, or maybe a little larger) have no good way to dissipate their heat and cold - the surface area is just too small.
5 - Finally, there is no insulation between the two boxes, and with a spacing of 3 mm there is no way you will ever provide it. So the heat separated by the TECs will immediately flow back to the cold side.
Those are the problems. What can you do?
1 - You must provide major insulation between the inner and outer boxes. An inch of foam is not too much.
2 - The hot side of each TEC must have a fan-cooled heatsink. The cold side also needs a heatsink, but that can be somewhat smaller than the hot side and may not need to have a fan. I would recommend the outer heat sink be physically connected to the outer box, while the inner heat sink is attached to (and supported by) the outer heatsink, and the inner box floats mechanically. Ideally, the layout of the inside of the unit will prevent food from being placed so as to block the circulation of cold air. The design of the inner heat sink (pins vs fins) probably goes in favor of fins, just for cost reasons. If you are unwilling to provide fans for the outer heatsinks, you must use larger heatsinks, and you must design the unit housing to prevent blockage of air circulation.
And for what it's worth, I'm with you on avoiding water cooling. On a portable unit that would be a nightmare.