Im learning as I'm ranting..
(bt1886 needs both a measurable black and white point, gamma 2.4s use (the theoretical argument for using it) is solely dependent on the black point, ...)
First: Write down your current setting parameters - as they are already quite good in principle. Dont loose them.
Next steps: Do a primaries/secondaries measurement and a saturation sweep (saturation of primaries and secondaries (0,25,50,75, .. percent)) and post the results (the graphs (saturation and CIE triangle) are mostly what we would want to look at), or interpret them on your own. If color performance turns out to be "uneven" in "certain ways" (color and hue slider would only address problems "uniformly" (across all color points)) - it can be a "device specific limitation" which then would not be addressable.
If your saturation sweep and color checker SG deltaE's turn out to be _just_ not optimal (as in fixable with the color and hue sliders
If you have a color and a hue slider you will first test them (certain settings) against the primaries/secondaries avg deltaE (one value, "can it be lowered even further"), then against the saturation curves (deltaE diagram), then against color checker SG values.
Also, If it is somehow possible that you could get the color of white line nearer to the 6500K target (without sacrificing _much_ greyscale performance (dE's)), go for it (as you have a gamma 2.35 and a 2.2 setting still to do...
). I realize that those are "fighting words", because after my slip up yesterday I tried influencing gamma with greyscale on my end (just to get to know the principle) and look at its effects on the gamma and the color of white curve, while at the same time trying to reign in greyscale - and I would describe it as "trying to wrangle a bull".
So good job already getting very near to the targets.