In other words, by not conceptualizing sensations of physical pain, do fully enlightened people still experience unpleasantness and simply not label it as such, or is the experience fundamentally without anything that an unenlightened mind would label as unpleasant, so that the degree of pain becomes like pitch in music?
It's said that enlightened person feels single pain (only physical), while not enlightened feels two pains (physical with mental pain).
SN 36.6, Sallatha Sutta:
The discerning person, learned, doesn't sense a (mental) feeling of pleasure or pain: This is the difference in skillfulness between the sage & the person run-of-the-mill.
An enlightened person feels the sensation corresponding to pain, but it does not feel as pain, rather as pure (informational) sensation.
My teacher taught me an exercise thanks to which I had this experience first-hand. It is called
kuem chok in Korean (I'm probably misspelling it badly).
kuem chok is performed by taking one of especially designed postures, "eagle posture" or "sleeping tiger" and others, which are inconvenient enough to generate bodily pain, but simple enough to maintain for a long time. The idea is to "go through" pain by exercising willpower, by not succumbing to the feeling of self-pity, until the experience of pain toggles and becomes something else.
After I was holding my posture for about 15 minutes, and had all my body shaking with pain, the sensation toggled, and instead of pain I felt heat and lightness. I could effortlessly maintain the posture and was surprised it felt difficult to me in the beginning.
This only happened once, and now the teacher moved on to a more difficult posture, "sleeping tiger", with which I was not able to switch yet. Needless to say, if I were you I would not try this at home without a guidance from qualified teacher, otherwise you may hurt yourself.
I am not referencing any suttas for this answer, it comes from my own experiences (I am not an arahat).
One who has obtained nirvana can choose whether or not to experience pain and can choose the level of suffering that is associated with that pleasure or pain. Having obtained nirvana an individual has the ability to reside beyond pleasure and pain and it also gives them complete understanding of pleasure and pain and things can not be understood without being experienced.
After some research, Sakalika Sutta: The Stone Sliver talks about how buddha cut his foot and endured the physical pain.
Excruciating were the bodily feelings that developed within him — painful, fierce, sharp, wracking, repellent, disagreeable — but he endured them mindful, alert, & unperturbed.
Although an Arahat would not have any mental pain arising with physical pain, that doesn't mean that everything would be just fine. Arahats could still find it burdensome, for example, in that it is constantly pulling on their attention and they wouldn't be able to do much as a result, so pain could still be quite debilitating in that the Arahat wouldn't be able to function by teaching, traveling, or other such things.
Of course enlightened people experience pain! Even the Buddha did. For example, in the Maha-parinibbbana Sutta, it is said of the Buddha:
When he had eaten Cunda's food, I heard,
With fortitude the deadly pains he bore.
From the sukara-maddava a sore
And dreadful sickness came upon the Lord.
But nature's pangs he endured. "Come, let us go
To Kusinara," was his dauntless word.
There is some evidence (and the exact source is escaping me now) that he was only free of physical discomfort when he entered into the base of cessation (i.e. nirodhasamāpatti). The Buddha even had to deal with a bad back on occasion (as in MN 53).
Of course, the experience of physical pain in not equal to the suffering of physical pain. While techniques do exist to deal with physical discomfort, only enlightenment liberates you from the anguish it causes.