Does every program
or disc [4K HDR] look as different as what you describe in comparison to the [4K SDR]? You may have to tweak-to-taste and write to one of those color/contrast menu presets in the TV or monitor's setup menu. Then you could switch over to whenever something looks too flat or tinty. Unless there's something else going on.
If you haven't yet tried, turn off any motion-smoothing and toggle the dynamic contrast on/off and see if that helps. Try to "force" the monitor not to "autoswitch" any settings whether it's receiving HDR, SDR, Dolby Vision, 709, 2020, sRGB, etc. –It's remotely possible that there's some signal-handling DSP that's being overwhelmed or is helping
It's times like this that I really wish all media players had a calibration "card" image that could be pushed to the TV or monitor simply by pushing a physical Test button on the device. And I do miss analog knobs for color calibration. Keep us posted. The screen shots do help. I really thought the ITU
was going to eliminate these issues.