Electronic – capacitance measurement circuit

• Use a star point for ground; with such small currents, the smallest impedance in the wire will distort your measurement.
• Terminate your coax cables properly; your coax cables are highly frequency dependend when you don't.
• You might want to use a high impedance amplifier near the device under test (DUT), like an active probe, FET-probe or something similar to drive the cables.
• When measuring such small signals, you might want to try to change the setup into a bridge https://en.wikipedia.org/wiki/Wheatstone_bridge

Okay, conceptually this is pretty easy, as I think you know.

A DDS chip from AD with sin/cos outputs, appropriate low pass filter, buffer amplifier. Apply a voltage much less than the bias voltage (but high enough to get good SNR) to the sample and measure the current multiplied by the sin and by the cos, low-pass filter the two results and calculate the real and imaginary components of the impedance from the (measured) voltage and (measured) current levels.

You should be able to add the bias voltage in the buffer amplifier, but you might want to capacitively couple the current input to keep the dynamic range of the mixer reasonable.

At 10MHz most precision analog multipliers are running out of steam, so I'd look at Gilbert cell mixers. Unfortunately the low-frequency and DC performance is seldom well specified.

Of course you could simply digitize the data at hundreds of MHz and digitally demodulate it with a fast FPGA, but that would be even more of a project.

The impedance of 0.156uF at 10MHz is only about 0.1 ohm, so the buffer should be able to handle tens of mA at 10MHz and your signal chain has to be happy with ~1mV total signal.

If you have access to a "lock-in amplifier" (the rack mount instrument), look at that to replace a chunk of the work. Same if you have a function generator with quadrature outputs.

I did something similar to characterize magnetic samples (there were some very special requirements) but the frequency had to be as close to DC as possible, so it was simply measured at low swept frequencies and curve-fit extrapolated to even lower frequencies (where there would be no SNR left).

It's not clear to me whether your model is primarily a series R-C or parallel R-C, of course the general measurement of Z gives you a complex number which could be applied to either model.

Your project also reminds me of some interesting work I did on conductivity cell measurements for dialysis water treatment. There were some heuristics involved.