First off, thanks for helpful advice and just to be clear, I'm not suggesting that building what I did is a good idea for all purposes. Clearly, what I have is suitable for only the smallest of DUT's and maybe not even good enough for that.
Now for the "still air" aspect of thermal transfer. "Heat transfer is classified into various mechanisms, such as thermal conduction, thermal convection, thermal radiation" (Wikipedia) Yeah, we all know that, right? However, unless there is a difference in temperature between regions, there is no transfer to worry over is there? In my situation, there is maybe 5.0 C max difference between ambient and what is needed to maintain constant temperature. As configured, I am heating the interior, not cooling.
"Still air" is a factor in convection, true, but a quick touch on the tip of your soldering iron will confirm that conduction is the king of heat transfer. In my little box, the Peltier transfers heat by conduction to a relatively massive AL plate. The plate radiates heat (or if the inner DUT box is in contact, conducts heat) into the box. That gets the air molecules moving faster and Brownian motion and entropy spread the heat around. I doubt that there is a significant convection component in the little box. Convection would be a major factor in a larger box and a stirring fan would be essential as TIN pointed out.
Consider the outer aluminum box. It is exposed to ambient, however it is also in direct contact (by four 4mm steel cap screws) with the metal plate that is the heat source for the box. In a way it could act like a "guard" being warmer than ambient and thereby lowering the differential temperature with the interior.
Now I have a controller that is sensitive to < .1C and can set hysteresis to as low as .1C, so now it is time for some experiments. I should have something to report by tomorrow.