Screens with profiles, or factory builtin profiles (to emulate sRGB for example), can indeed sometimes have ugly nonlinearities that can make images appear to have problems when there are none, I've seen that. However I move between several computers, some without screen profiles, and I also know the profile quite well of my own screen (shaper profile designed with Argyll) so there was no such problem here.
The problem was as described in the last post, due to that there was a poor fit the LUT had to stretch a lot, and then the results become very white-balance dependent. If the white balance is not exactly as the profile expects the LUT can drag the wrong colors which happened in this case.
The next problem is why there was a poor fit and I'm still investigating that. I think there was some measurement problem, either in my XYZ reference measurements or in the test target photo, or both. I'm starting to suspect that high saturation patches as I have on my test target may not be a good idea when you shoot a target as it seems like the impact of small measurement errors becomes large, while lower saturation targets are much more robust. This is still just a theory though which I need to verify.