and how are you measuring this "0.001" degC ?
From the pictures in the links I provided, you may notice a sensor located pretty much in the center of the box. There is a fair bit of air flow inside and it should have little dead volume. The sensor is a platinum RTD that is read through my HP 34401A bench meter. The PC reads the meter and performs the conversion. The conversion is supplied by the manufacture. Keep in mind that my goal is not to have absolute accuracy but rather short term stability, followed by repeatability and then accuracy. If you have specific questions, feel free to ask.
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I forgot to mention that normally when I am running experiments where I need stable temperatures, I will normally have other test equipment near by controlling and collecting data. All of that equipment also creates air currents and heat which can cause problems with my measurements. I expect OP may have a similar problem.
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A link showing some of the basic differences between sensors:
https://www.omega.co.uk/temperature/z/thermocouple-rtd.html
OP needs only a square cm for his diode, so his task is easy. I've put a 10 cm foam on top, so above the heatsink there's only a tiny room of air. The diode and sensor are glued to the heatsink, so most of the heat will be taken from that. I guess the air might be a tiny bit cooler.
If I where to scale this setup to a larger box, I would probably have a box in a box where the inner box would be surrounded by water. The outerbox would then have styrofoam isolation on the outside. Then have maybe the outerbox made of alu or something with heaters to keep the water at the requested temperature. Maybe with dual heaters, one water boiler 2300W for fast heating, and one much less powerful to keep it at the target temperature. Maybe throw in a waterpump as well, pumping water trough copperpipes mounted on a isolated alu lid. Inside the box the temperature will eventually be the same as the water temperature.
In the inner box, there also should be a fan to move the air around.
Maybe one day I'll have a purpose for this idea.
I assume that the peltier aficionados know the following vids/channel:
https://youtu.be/aSGwaQWA05UI really like the stuff they're doing.