The cold junction compensation is only really caring about the thermo-power from the virtual reference temperature (e.g. 0 C) and the temperature of the actual junction. Just imaging using the simulated cold junction to also measure an actual thermocouple at 0 C in parallel.
One could either use the simulation or the actual TC to 0 C.
Really important is only the thermal EMF around the cold junction temperatures, so around room temperature. The part in between is just to set the actual offset.
The piece of copper is really huge. Thermal equalization scales with the distance square, so smaller can really be faster. It should still be much faster than the water version. Symmetry can reduce the effective size to about half.