Probably Brienne. She's an essential part of Jaime's arc but other than that I don't see her having a real purpose in the story. She's really not all that interesting when she's with Jaime, let ony without him.
Obviously grrm doesn't really agree with you, since he gave her many povs for herself.
I can't with people seeing "no real purpose" in characters. The story isn't finished yet, so unless you're grrm or talked directly to him, how do you know? It's fine not to like characters, but it has nothing to do with purpose, just say you don't like them.
As for who will die first, I don't know really. I don't think Jaime has more left to do than Brienne, I think they're about the same level.
I am thinking Brienne will sacrifice herself for Jamie's life. Jamie after all came back and saved her life, so she has a life debt towards Jamie. Though I am not sure how Lady Stoneheart would treat with Jamie. Especially if the popular notion is that Jamie will kill Cersei which means he would traveling back south and that takes time.
I see this all the time, but personally I don't think this is the way the story has been set up. There's a parallel between Jaime and Brienne, and the fact Brienne is now facing the "too many oaths" problem, in a similar way Jaime did before her. This whole story is about Brienne as much as it is about Jaime, in fact when it comes to LS it's more about Brienne as she's the one faced with the conundrum.