If you take knowledge that gets things done as one form, and knowledge that captures data as a second, and your third entry is messy, emotional and mysterious, to my mind, you are setting up a trip around the one of the three elemental configurations that maps directly onto Lacan's four discourses, which we can map further onto the powers of the sphinx. The point is not to be abstruse, but to have any model of knowledge that actually got used and included similar divisions.
There is a point in noting the greater similarity to part of a set of four, instead of looking at this as a Fixed/Mutable/Cardinal configuration (equivalently Real/Ideal/Symbolic), other than the difficulty of fitting the latter. I think, given your examples, that you are vaulting over a more obvious form of knowledge that includes aesthetic decision-making, the ways we 'read' people or situations (even if these are just recursive 'theory of mind' predictions) and the selecting out of which experiences constitute stories with significance. Quale, familiarity, suffering and numinousness, etc. are on a layer below that after individual rational psychology, intersubjective 'chaos' and the irreducibilities of mutual social manipulations have been removed.
So I might suggest that this third/fourth form of knowledge is the hysterical factor to a world that is first analyzed in terms of effects and contents, (and then maybe philologically/deconstructively/aesthetically.) Since 'hysterical' is provocative, I want a synonym. I am going to use both words, but alternately. Hopefully one will be a foil against the annoying nature of the other.
If the Master discourse is Will and the University discourse is Knowledge, the Analyst discourse is clearly Daring, and the Hysterical discourse is left to be the enigmatic Silence.
The Hysteric's discourse does what is necessary for knowledge that cannot be captured or used to be embodied, and not lost. Silence is a posture that allows the universe to be solid, with all of the movement introduced by the three active principles grounded out so that the three other sorts of power have a stable position from which to arise, and back to which to return.
So what use does science make of Silence? I think that one form of Silence is embodied in the structure of emergent principles that allows for the separation of disciplines. By solidifying quantum motion, chemistry can have a domain of study. Orbitals are not a lie, but they are surely less than real. This knowledge that what goes on down there should only be considered when the normal way of looking at things fails is a proper and productive Hysterical repression.
The layering also allows each different science to have its own paradigm and its own character. Like the way someone is your friend only because you know them in a given way, you can like Chemistry and dislike Physics even though Chemistry is Physics. So you can look at Silence as the thing that holds a paradigm in place. Paradigmatic function is about questions that one does not ask, because we have decided, quite properly, but kind of oddly, not to ask those for a while, while pretending to keep an open mind.
Your notion of qualia is also intimately tied up, in my mind, with the way in which emergent properties like 'temperature' are very real, but don't exist independently. Temperature has in common with redness that it represents a statistical convention that suppresses detail to good effect.
Looking at how these principles are part of the process of science might indicate how science itself should incorporate them into its explanations.
These are aspects of the process of scientific decision-making itself, which requires its own separate model so that it can bounce back and forth against the theories of what has been observed and predicted.
If you do not have a theory of observation its requirements and its limitations, even after factoring out biases due to social and intellectual factors, your model of the observed phenomena will have systematic flaws that cannot be removed by any process.