The capacitance is dominant at low frequencies and minimal transmission line effects. So measure S11 with the other DUT terminal open and you are measuring C11||(C12+C22); "+" means in series, "||" means in parallel. Now short the terminal to measure C11||C12. If you assume C22=C11 you can compute C12, or reverse the DUT to obtain a 2nd measurement to solve for all three unknowns.
Now you need to find the 1st resonant frequency (series resonance). You can do this from S21 of a series measurement, but better to put the DUT in shunt and measure the frequency and depth of the notch. From the resonant frequency you can determine L11+L22, and from the depth of the notch you can determine R11+R22. Of course technically the R and G can be obtained from a DC measurement, but they are a function of frequency so you could get different results, and G is so small you could not measure R.
Then you would compare the lumped model response to measurements and tweak it more.
So really it's a matter of measuring at low, high, and resonant frequencies, and terminating the output with opens and shorts. From this you can extract values for the lumped elements. You can even lift the DUT away from the ground plane to minimize C11 and C22.
If you read up on small signal model extraction they will detail some of the methods.