If this device measures capacitance by measuring the current at a given frequency, the current drops as the frequency goes down. This is due to capacitive reactance being proportional to frequency. (X = 2*pi*f*C) At low frequency, the current is very small and hard to measure reliably.
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.
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If it were me , I would not only measure C & tan delta but ESR as well and test to 10MHz.
You can use a scope rear sweep signal to drive an FM generator, but 50 Ohm is not the best source.
If you cant scrape a simple AC current source design to generate cap impedance as the output voltage, use a voltage source (ie. darlington emitter follower to drive the cap with an adjustment for DC offset +/-2 V and monitor current with a 1 Ohm shunt. The scope can show the DC bias on one channel and ac coupled envelope synced to scope sweep. This is a quick & dirty method. XY mode of V vs I is another method.