I am curious as to why these problems appear, in such cases I have been known to offer suggestions and thought-experiments that can be impractical, investigate asymptotic behaviour etc.
Would a multi-spectral camera (if one suitable were available) coupled to a multi-spectral workflow and printer solve the problem? Or is it more philosophical about the idea that one can do a 2-d "image" of a scene, move it to another time another place and expect to recreate the same perceptual response?
One problem is the brain's evening-out-effect, most clearly demonstrated with mixed artificial light. If you photograph that and want to have a realistic representation you will feel that you need multiple white balances, to bring the lights closer together as it was experienced on scene.
The simplest way to see on this is that the brain fully compensates, ie "auto white balance" so the brightest spot is seen as pure white, and to model this behavior one can shoot a gray card and the white balance with a color picker. However it's more correct to say that the brain brings the "neutral" color closer to pure white but not all the way. In arctic dusk light the "white" is experienced as very far from white so this model will not yield a realistic result.
As far as I know there are no models made of the brain's "white balance" function, so even if you have a full spectral recording you don't have a model that can translate that into how the eye experienced it. I do think it would be possible to make such a model based on experiments. Another aspect is that possibly the color response changes more drastically (like it does in low light) than just a white balance difference so you would need a totally separate DCP/ICC for more extreme light. The standard DCPs are typically made for 5500 and 6500K. I don't really know how well those translate into extreme light. Probably not perfectly, but my guess is that it's not too bad either, if you can secure the white balance then you have come a far way.
The eye/brain has other "evening out" effects to, for example that you much easier can see past reflections on a window (it's about depth seeing) than in a photograph, or that you don't see haze as disturbing in real life as in a photograph, which means that you may use a polarizer to reduce reflections and slightly reduce blue channel content on distant object to get more realism.
And then we have all other sorts of lighting conditions, low light, narrow band atmospheric phenomena like northern lights etc.
(And we have print paper and viewing conditions which affect the experience, but I think people have a tendency to emphasize these issues way too much compared to color differences occurring in the post-processing step. If the paper is warm-white or cool-white has a much smaller impact on the realism of an arctic winter photograph than choosing appropriate white balance in post-processing. Those are fine-tuning problems.)
About here people get bored and think "it will never be accurate, so why care?" and just make something they find pleasing, not caring about realism at all. I'm fine with that, but some of us find this interesting and I prefer being able to talk about the subject without getting into meta discussions about what photography really should be about
. Sometimes I use the camera response creatively, long exposures in low light are very different from eye response for example, and sometimes I want a realistic rendering of the eye's experience at the scene, such as my arctic winter example. I think it becomes more rewarding to capture extra-ordinary light if it can be realistically reproduced.
In one aspect capturing extra-ordinary light on film was more rewarding than it is on digital. Film did not necessarily render it realistically, it have less potential to do so than digital, but the color conversion was the same so something extra-ordinary in real life became something extra-ordinary in the picture. With digital you just play around with contrast and saturation so everything looks extra-ordinary in the same way. You can be disciplined and limit your manipulation, but in extreme light I find that a challenge arise to find a realistic starting point.
Even in less extreme light I'd be glad if I could have a more realistic representation of colors, I've been shooting fall colors recently with my medium format system and I just did not manage to replicate a color response that seemed true to the scene. Either too brown or too green, couldn't find a proper mixture of yellow and greens. I'm not sure if it was my memory failing me or something else. In the end I chose something that looked pleasing, although I'd prefer the original if I could have replicated it. So I have become interested in what techniques there are to reproduce realistic colors. Due to the brain's "evening out" effects maybe the most true-to-the-original-scene impression is not given by the most accurate colors, there may be some psychovisual effects involved.