Hi Mike,
I do appreciate your taking the trouble to do some of the numbers to find out what will fly here, and also posting the link which i found which is very informative. Now what i propose is that you redo the calculation one more time but this time take into account the natural rise in temperature of the hot side once the thing is plugged in. Once the unit is plugged in the first thing that happens is the hot side starts getting hot as it pumps the heat through the device from the 'cold' side. Now it's unfortunate but that means the hot side gets hotter than the ambient and that means the hot side of the Peltier device itself gets hotter than ambient. The downside to this is that the temperature differential has to be more than simply the ambient minus the inside colder temperature because the temperature differential is not the outside air temp minus the inside air temp, it's the hot side surface temp minus the cold side surface temp. Add to that the problem of heat sink thermal resistance and we're suddenly working with a less efficient system.
My only suggestion here can be to build a smaller prototype and test it, then scale up from there. Build a small unit with one Peltier device and use two nice size heat sinks and two fans. One fan on the inside and one large fan on the outside. Start with a small styrofoam cooler (cheap) and cut a hole in the top to fit the Peltier and go from there. Measure inside and outside temperatures. This would also allow us to test the time it takes to get the mass inside the box down to the target temperature, using a mass with proportionally sized specific heat capacity or at least something to simulate the true object that will be inside the box in the final project.
As Ron pointed out, there seems to be more to this than we have considered so far (such as the heat of the hot heatsink and not just the ambient temperature) because i dont think those commercial units work as well as we would like. Lucky for us here though is that we dont need a huge ambient to inside temperature differential