The facial injury could be a clue, but perhaps to Brienne's heritage not her story arc. I think this because I also have a crazy theory that she'll heal with minimal to no scarring. I was thrilled that Brienne had no scars from the bear in season 8 because it seemed to confirm my theory. It really isn't hard to put special fx scars on an actor and her neck was 100% healed in season 4.
My crazy theory is that GRRM based the Targaryens on the Old Ones in Tanith Lee's The Birthgrave Trilogy. If I'm right, or close, then Brienne's facial injury is a clue to her heritage. If not, GRRM was getting unnecessarily sadistic in AFFC.
The Birthgrave begins when a white-haired woman wakes from "death" in the heart of a dormant volcano. Karrakaz is a member of the albino Old Ones, despots who ruled over the lesser peoples with superior technology, practiced incest and went mad. They do not scar and heal rapidly even from deadly injuries. Somehow, they wiped themselves out and the lone survivor took refuge in the volcano (I don't remember how they all died). The protagonist spends some time among barbarian nomads who greatly resemble the Dothraki.
When Brienne appraises herself in the mirror in AFFC, she does not note any scars from the bear pit. She's intensely self-conscious, so if she had disfiguring scars along her neck and torso, she would add that to the list of things she hates about herself. After she's bitten, she is told it will scar and seems shocked at the thought, like she has no scars at this point in the story. This is much like the protagonist of book 2, Shadowfire (aka Vazkor, Son of Vazkor), whose skin never scarred because his mother is not human. The link below gives a great overview of this book if you're interested.
https://jamesdavisnicoll.com/review/whatever-happened-to-the-baby-abandoned-in-the-birthgrave
Here is the opening passage from Shadowfire.
Book One: Part I: The Krarl
"One summer when I was nine years old, a snake bit me in the thigh. I remember very little of what followed, only being mad with heat and tossing about to escape it as if my flesh were on fire, while time passed in patches. And then it was over and I was better, and running on the green slopes again among the tall white stones that grew there like trees. I learned after that I should have died from the snake’s venom. My body turned gray and blue and yellow from it; a pleasant sight I must indeed have been. Yet I did not die, and even the bite left no scar."
https://www.amazon.com/gp/product/B00S75RXJQ?ref_=dbs_m_mng_rwt_calw_1&storeType=ebooks