The notebook has received quite some love by @nick and I, no underlying model changes but it is overall better!
We will actually submit it to the John Hunter Excellence in Plotting Contest - 2020. Here is the almost final page: https://colour-science.github.io/gamut-mapping-ramblings/ Once this is done I will tag the repo as first like @jedsmith did on his.
When comparing both the RGB Saturation and HSV Control models I noticed a cyclic pattern in the way they differ: the HSV Control model tends to clump the intermediary colours toward the primary and secondary ones which I assumed could be solved if not entirely, at least partially with the Hue Controls.
RGB Saturation - Full Effect - No Threshold
HSV Control - Full Effect - No Threshold
Difference * 4:
With that in mind, I tried to naively brute force the required Hue Offsets to reduce the error between the two models. Interestingly enough and even though I did not give a chance to the optimisation to finish, here is the resulting Hue Lookup with abscissa representing the Hue Angle and ordinate the Hue Offset:
And normalised with the location of the primary and secondary colours, so that it is easier to appreciate the underlying signal:
If you mentally get rid of the noise, there is a triangle/sine wave hiding here.
And finally the difference with the two models but this time with the noisy signal applied to the HSV Control one:
Difference * 4:
Quite cool because it confirms my gut feeling that the HSV Control model should be able to achieve results in the ballpark of the RGB Saturation one. I will come up with clean offsets in the coming days. Of interest for you @matthias.scharfenber 
Cheers,
Thomas