As bizarre as it seems, this detail was actually inspired by Inuit accounts of the fate of the Franklin Expedition. From a 2018 interview with the creators Soo Hugh & David Kajganich:
[T]he image that really gets under your skin is the sight of Edward; dead-eyed but still alive, with chains strung through his face. It's a rich, mysterious moment that Hugh explains was pulled from actual Inuit testimony.
"It's interesting, we've gotten so many social media questions about that imagery. And we debated how much to reveal, but I think I want to put point on it just because people have been so mesmerized by it. That imagery actually came from a testimony about discovering one of the men from the expedition, his corpse, with chains around his face, linked to his ear. I don't remember when we found it in the writer's room, but I remember all of us just sitting there quiet, I think each of us were imagining that in our heads and we knew we had to bring that into the story. We wanted to bring as much Inuit testimony into the story as possible, but especially that one, which was so indelibly haunting and full of ambiguity."
"It was a mother and a son, an Inuit woman and her son, who found that particular tent with a body in it that they had said he was rigged up with a set of chains that were in his face and that if you pulled one of them, it pulled his head up slowly. They had no explanation for it, and the people interviewing them had no explanation for it."
It appears that the creators found an account reported by William Henry Gilder, who was second-in-command on a 1878–1880 expedition to search for relics of the Franklin expedition, and who later wrote a book recording what he heard and saw. Here are the relevant passages (bolding mine):
At the next camp, which was our last stopping-place on the main-land, we met an old woman named Tooktoocheer, widow of Pooyetah, who was among the first to visit the boat place we saw a few days ago. We were somewhat disappointed in her as a witness, for she was so old that her memory was at fault, and she would wander about to different places and relate circumstances without explanation. Her son, who was present at the interview, was a lad of about twelve years when he visited the boat place with his parents, and retained a vivid recollection of the place. His testimony, therefore, proved to be what we had hoped of his mother's. ... I give the interview with Tooktoocheer and her son as I recorded it in my note-book at the time, so that each reader may draw his own conclusions. Some of the statements will undoubtedly appear strange, but in the main they are perfectly intelligible and exceedingly interesting.
[Ogzeuckjeuwock, Tooktocheer's son] said the appearance of the bones led the Inuits to the opinion that the white men had been eating each other. What little flesh was still on the bones was very fresh ; one body had all the flesh on. ... One body — the one with flesh on — had a gold chain fastened to gold ear-rings, and a gold hunting-case watch with engine-turned engraving attached to the chain, and hanging down about the waist. He said when he pulled the chain it pulled the head up by the ears.
His statement in reference to one of the deceased wearing a watch by a chain attached to his ears appears strange, but I give the statement as he made it. The chain may in some way have become attached to the ears, or, ridiculous as the story sounds, there may have been some eccentric person in the party who wore his watch in that way, and if such should prove to be the case, this would certainly identify him beyond doubt.
The creators seem to have embellished this a bit for visual effect, but even at the time this was viewed as a bizarre thing to have happened.