I did a fairly detailed VLF antenna and receiver design project a few years ago It was for use in the few 10’s of Hz to a few hundred kHz range, This was for reception of various naturally occurring and manmade over-the-air signals, and required performance down to the background noise floor.
So, it isn’t exactly what you are looking for, never the less there were some general rules of thumb that emerged from my analysis and design that you may find helpful.
1 – A LOW/ZERO input impedance amplifier connected to the pickup loop is the “natural” topology for flat frequency response. fLow is when xL of the loop equals Rin. fHigh is determined by the self resonant freq of the loop. Response is flat in between.
2 - Best system SNR occurs with a ONE turn pickup loop of whatever your maximum diameter is. This may seem somewhat counter intuitive (it seemed that way to me at the time), However, it is not always practical to use a 1 turn loop, as the amplifier may not have enough open loop gain to achieve desired output voltage. I ended up with a 9 turn loop, which had a low enough system SNR to still “see” natural background.
3 – The LOW or ZERO input impedance amplifier or op-amp should be chosen for the lowest equivalent VOLTAGE noise (en) for best low noise performance. Current noise has very little impact. In my design, zero input impedance meant connecting the loop to the inverting input of a closed loop op-amp. Another design I have seen is a Stanford VLF receiver using a discreet low input impedance design and an input transformer, but details of their design were sketchy
4- “Gain” is best calculated as a transfer function, i.e., Volts/Tesla. This gain is a function of the loop diam, the number of turns, as well as the loop inductance and op-amp feedback resistor
I had to put together a good sized spread sheet that calculated all of the above to arrive at a final design.
For your design, it is not clear how to calculate the effect of the shield. But the general guidelines above should still apply.