So the Jaime redemption debate has been going for goodness knows long. Some say he most definitely will. Some say he most definitely won’t.
GRRM’s opinion on this is that he wants the reader to decide for themselves. He doesn’t want to spell it out for the reader whether or not Jaime has redeemed himself (even though he does lean towards redemption always being a possibility.)
So when considering Jaime’s ending, it’s not worth fixing it around whether or not Jaime has been redeemed. Instead, it is worth focussing it around whether or not Jaime’s redemption can be open to interpretation. There needs to be ambiguity.
I would argue for this ambiguity, it is necessary for Jaime to be at least somewhat reformed. Whereas redemption is about making up for sins of the past, I think being reformed is about who you are in the present. Seeing as the doubt as to Jaime’s redemption come from his past villainous actions, there isn’t much ambiguity as to whether he has been redeemed if he continues committing evil acts. So in terms of who Jaime is going to be at the end, it is a man who is doing better things than he did at the beginning. If he was ‘evil’ in the past and ‘evil’ in the present, then the question of redemption really isn’t up open for debate. There has to be progress, there has to be the choice for fans to decide he has been redeemed.
As for what ending he will have, happy, tragic, in the middle, well all of them are possible.
A happier ending can have the audience asking, “did Jaime redeem himself enough to justify that ending”. Jaime can have a chance for a new life and still carry the wrongs of the past with him.
A tragic ending, it is the opposite. The fans can debate with whether or not they want to see him punished as much as they did when he first pushed Bran from the window.
A middling ending, remembered for a heroic death, goes into hiding, joins the Night’s Watch, suits the ambiguity of whether or nor he is redeemed.
I will say this, I think GRRM has warmer feelings for Jaime than most of his readers. He was the one to raise the question with a reader about what they would do if they had to kill a child to protect their own family, which is the most sympathetic reading of Jaime’s actions in that chapter. And that; mixed with GRRM’s leanings towards the possibility of redemption, will colour the tone of Jaime’s endgame.