I soldered the servo circuit on a prototype board. The transfer function of the soldered board is matching very well with the analytical calculation. We found a few more corrections required in the circuit relating to the voltage offset compensation resistors. Then I did a complete noise analysis of the circuit using SR785. Attached are the results. First I measured the transfer function of each individual stage by sourcing the input of the circuit and taking the ratio of the output to the input of each stage separately. Then, I used 2 different methods to calculate input referred noise at different stages of the circuit (which are soldered together).
Method 1: I connected a 50 Ohm terminator at the input of the stage and measured output noise of the stage. Then I divided it by the transfer function of the stage.
Method 2: I only measured output noise at the output of each stage. And calculated the input referred noise by dividing the output noise by transfer function and subtracting the excess noise from the previous stage at its input.
In the process, I have written a module which can be used by others too to fit transfer function data and calculate input referred noise in general using SR785. I would put this code on github soon.
It seems that as the LISO analysis shows, most of the noise below 20-30 Hz is due to the amplification of input current noise of the first stage amplifier. Next steps are to figure out a better opamp if any to reduce noise in this region. We are well below the shot noise for 2mW 1550 nm laser above 30 Hz.