I wanted to take a direct spectrum noise measurement of the circuit. However, if I have simply hooked the output of the circuit to SR785, I wouldn't know if the noise is coming due to temperature fluctuations or due to circuit noise itself. So I made a quick subtraction circuit of unity gain with an OP27 and subtracted different pairs of channels and then took their spectrum using SR785.
Method:

First I shorted the input to the subtraction circuit and measured the spectrum. This gave me an estimate of output noise due to this additional circuit and SR785 measurement.

I assumed the gain of the subtraction circuit is just 1 so, the output noise of the circuit under test wasn't assumed to get amplified from subtraction circuit.

Then I took measurements of the difference of signals for each pair from CH0, CH1, and CH3 in both permutations. All measured spectrums are plotted in the first plot.

I used both permutations to nullify any effect of different gains in positive and negative inputs of the subtraction circuit due to slightly different resistor values.

Then I averaged the spectrum from the 2 permutations of difference and subtracted output noise of the subtraction circuit measured earlier in quadrature. These are plotted in 2nd plot.

Then for the Circuit 1 which was insulated from the environment, assuming the two circuits have identical noise, I divided by sqrt(2) to get individual channels output noise.

I subtracted this insulated individual channels output noise from the difference noise with 'open to environment' channel in quadrature to get an estimate of noise when the circuit is open to the environment.

The individual channel output noises are plotted in Plot 3.
Conclusions:

There is a high peak at 60 Hz because of AC frequency. I am not sure about the origin of the 7 Hz peak. Other peaks look like reflections during the measurement.

There is about a factor of 2 improvement by insulating the box.

The circuit is designed for 1.625 mV/mK transduction.

Since EPICS channels are read at 16 Hz rate, we should be interested in the 016 Hz bandwidth. In this bandwidth, voltage output noise for the insulated channel is 0.79 mVrms and 'open to environment' channel is 1.74 mVrms.

So the noise wouldn't harm the designed 1mK sensitivity. In practice, even if these estimates are bad, the circuit should be ok for our needs.

So the only thing that can endanger our precision really would be temperature drift of circuit components on which we have already invested enough money and time during design.

So agreeing with Andrew, I'm going to go ahead and install these new circuits and directly see their performance towards improving BN spectrum as we think the Vacuum Can temperature is oscillating more than lab temperature due to poor circuits presently employed.
