One small problem...

It looks like you've just built yourself a rather odd thermos flask.

I'm no expert, but it strikes me that you won't affect on the internal temperature much once you evacuate the conductor (air) from within the container.

Therefore, I suggest a great simplification - leave the temperature probe in the flask until the correct temperature is reached, then withdraw the probe, seal the top and then evacuate the air. The end temperature will not vary much during this process.

The advantage will be fewer holes to plug, truer temperature readings and generally follows the KISS principle better.

BTW, how brittle will the metal container be at these low temperatures? Is this a safety or otherwise concern?

One final point worthy of consideration - the dry ice will create copious amounts of a mist-like low-lying cloud, which will hamper finding and reading valves and gauges unless this is taken into account (e.g. raise the unit on a stool, lengthen the hose between unit and gauge, add a fan to the mix, ...)