Could Brienne ever truly be a knight?
For example, if another knight decided they wanted to knight Brienne, and did so publicly, is that end of story? Brienne is a knight much to the chagrin of her haters?
Officially, yes. Any knight can make a knight. So when the local lord anoints his failson with rhe seven oils, yes, he's a knight and afforded all the privileges thereof. Same goes with Brienne.
Unofficially, Brienne's knighthood only extends as far as people are willing to accept it. She's be shut out and denied from those privileges unless she forces the issue. Games masters would find technicalities to deny her participation in melees, lords would refuse to swear her to service, and so on and so forth. Social stigma is a powerful thing, and exclusive castes try to do everything they can to maintain exclusivity - lest the allure wear off their exclusive caste. One of the most advantageous assets of an exclusive caste is the right to deny it to those who are considered unworthy. So I would imagine that knights would fight against Brienne with every unofficial trick and technique to deny her until she is dead or discredited enough to annul the knighthood.
Thanks for the question, Anon.
SomethingLikeALawyer, Hand of the King