I was looking at The Journals of William A. Lindsey, a reprint of an Indiana doctor's journal with several 1830s procedures. As usual, he doesn't describe what the patient actually does to cope with the pain, although he acknowledges it:
p. 77: I was fully 2 hours in dilating the uterus, and fully another hour in detaching the placenta... The operation was attended by excruciating suffering to the patient, and it felt sometimes fearful that she would be unable to bear up under it, and I was several times compelled to desist a short time owing to the exhausted and sinking exacerbations induced by the operation.There's also the question of whose responsibility was it to deal with the pain? In modern life, I can't ever recall a doctor instructing me how to cope with pain. The dentist comes with the needle: "Just a little pinch," but never says "Be prepared to grip the arms of the chair" or "You can whine and whimper if you want,"--that's supposed to be my responsibility to initiate.p.66: Having made known to him... the necesity of an operation, I shall never forget the earnestness with which he emplored me not to operate on him. and he so far worked on my feelings & sympathies that it was with much difficulty that I could man my self up to the operating point; I had a son at home near his size & age, and could not help anticipating what my feelings would have been, had this unfortunate boy been my own son thus emploringly pleading with me not to operate on him... But I was much encouraged in the operation which I so reluctently commenced, by the manly resignation & fortitude of my patient, when he found the operation was commenced & would be performed.
So would the doctor say, "Bite down on this while I remove the bullet?" or was the coping method up to the patient?
If the soldier was more familiar with surgery at home, he might remember what grandpa did when he had to have that cancer removed, or whatever. If the soldier was basing his actions on military life, and knew of the tradition of biting a bullet during punishment, he might choose that.
All just speculation, of course... Just different ways of considering the issue.
The biting of bullets by soldiers during corporal punishments seems to be the most common thing discussed in the early 19th century, as far as bullets go, though it's hard to say if the commonness was exaggerated because it's something "everybody knows" they did, or if it's something they actually often did.
However, it occurs to me that during punishment, the smallness of a bullet would be an advantage, because if the goal is to show how brave and stoic you are, biting on anything large enough to be visible would advertise that you needed help to keep from screaming. Also, you'd be standing or lying face-down, thus less apt to swallow it (though surgeries or minor procedures without anaesthesia were sometimes done seated as well).
The following is from the Sketches of Philosophy of Life by Sir T.C. Morgan, M.D., London, 1819.
Hank TrentBy the same principle may be explained the very singular fact, that great mental excitement suspends or supersedes the impressions of bodily pain. It elucidates, also, the relief which soldiers seek under punishment, from forcibly biting a bullet. In the latter case, the whole force of volition is expended in exciting the maxillary muscles; and by thus rendering the brain, in some degree, a centre of fluxion, it diminishes the irritability of the tissues, which are torn by the lash of the executioner.