The fact that neuroscientists might one day deliver an explanation of the soft problem of consciousness (i.e understanding exactly how the brain works and the exact correlation of every state and process to the corresponding subjective experience) will not bring any of us one iota closer to understanding the hard problem. In this case, the fact that one does not recognise there is such a problem implies ignorance more than anything else.
History is filled with examples of problems thought to be unsolvable until they weren't. Many things were in a mysterious realm of non-explanation before being explained by science.
It could be that the current problem is poorly defined or the wrong questions are being asked, but nothing suggests it's an unsolvable problem 'using any scientific method'. You can also easily see the material nature of consciousness by looking out how physical things affect it (brain tumors, LSD, alzheimer's).
Particle physicists are also continuing to work on the building blocks of matter. You can use science to figure out the nature of reality in the state that it really is, not just how it appears to us in our mental models of it. The map is not the territory.
First, of all, it should not surprise you that there are limits to what can be achieved with reason or knowing how things work: Within the boundary of what can be thought about there are already problems that can be proven to be unsolvable, such as finding an algorithm that can tell whether a program will halt or not (undecidable problems), or finding a formal system of rules that is both complete and consistent (Gödel's incompleteness theorem).
Now regarding physics - you said it yourself: The map is not the territory. Physics is nothing more than an ever improving map. It will never be the territory. The mental models you speak of are exactly the ever changing map which attempts to fit reality (which is what exists prior to giving it a name) to concepts which make sense to us. It is crucial to understand that our concepts/thought about reality will only ever be a subset of reality. Equations are nothing more than a description - they are not the thing in itself. Consider the question Stephen Hawking phrased: "What is it that breathes fire into the equations and makes a universe for them to describe?"
The last thing is understanding the difference between the soft problem and the hard problem of consciousness. These two problems are not on the same level at all. The soft problem can be successfully used as the "God of the gaps" type argument and therefore I could in principle agree that it can be eventually be resolved (though I might argue regarding the time frame required to solve it). The soft problem is "simply" fully understanding the functionality of the brain, part of which includes an explanation of which physical processes in the brain are followed by which experiences. The hard problem asks a completely different question: how does a physical process give rise to a subjective experience? Saying that the subjective experience is just some illusion while the physical process is real is silly considering your only access to reality is via your subjective experience, followed by your theories on what that experience means (including the thought that there are such things as physical objects).
Now the reason the hard problem of consciousness appears solvable to many people comes from the mistaken view that consciousness is an emergent phenomena of physical processes, that the "subjective" is emergent from the "objective". These terms are of course necessary for science to exist, but as many wise people who have meditated long and hard on this matter have suggested, the nature of consciousness exists prior to concepts, prior to the division of reality by the mind into subjective and objective and therefore prior to (and so beyond the scope of) scientific inquiry.
The halting problem is a good example that I'd have to think more about. My initial feeling is that while the proof shows that you can't write an general algorithm to determine if a system halts you can still understand why that's the case from the proof itself. My claim isn't that understanding things allows you to do everything, but to find out what can be done. You're right though that knowing more won't let you violate the laws of physics or computability - it just lets you learn what the laws are.
For the physics point I think it's possible that the territory is math and that's why mathematical equations do such a good job of explaining it. I read about this idea in Max Tegmark's book: https://www.amazon.com/dp/B00DXKJ2DA/ - in that case we could be discovering pieces of it (though our syntax and such is still just a map to it). Until we have a grand unifying theory it seems worthwhile to keep working on the map. Maybe I'm just optimistic about this.
You lose me in the next section though - it seems like we agree on the soft problem, but I don't understand your classification of the hard problem.
Why is it difficult to think of a physical process leading to subjective experience? Natural selection towards a general problem solving organ that provides feedback etc. Visual inputs, audio, the ability to make predictions and act based on those - subjective experience seems like a natural outcome of all of that?
I think you and I probably agree that 'emergent phenomena' is mostly hand-wavy nonsense (http://lesswrong.com/lw/iv/the_futility_of_emergence/).
> "but as many wise people who have meditated long and hard on this matter have suggested, the nature of consciousness exists prior to concepts, prior to the division of reality by the mind into subjective and objective and therefore prior to (and so beyond the scope of) scientific inquiry."
I don't understand what leads to this conclusion - your last sentence seems to imply dualism which I think is generally pretty weak since poking someone in the brain, a brain tumor's affect on consciousness, and LSD's affect on it provide decent evidence that it's material. I don't think this is a real problem, it feels like one of those that disappears once we understand how things actually work.
