Do we really know that much about Bronn and what he’ll do from one moment to the next? He seems motivated more by survival and riches than he does by knighthoods and empty promises made by those that are so quick to renege on their word simply because they see Bronn as someone that is less than them. Bronn no doubt thinks very little of their opinion, he wants only what he was promised and nothing else. For a sellsword he’s just as greedy as he needs to be and he expects people to live up to their word. One thing about Bronn is that he does have a sense of justice about him, but it’s used in a purely personal manner. So long as he gets justice and is dealt fairly the rest of the world can do what it likes.
That’s why Bronn is seen as such an endearing character really. It’s not so much that he’s just out for himself, which he is, but he wants to be treated fairly for his services. After all, switching over to the side of the Lannisters has not made him any friends, not even among his allies. But Bronn is a survivor and knows how to get by, even when he’s firmly set within the Lion’s den where any one of them would gladly stick a dagger in his back.
Why he saved Jamie is purely based on the fact that you can’t collect your due from a dead man, and he no doubt wants to avoid Cersei at all costs since she seems like she would gladly set the Mountain on him as payment for his services. At this point Bronn really has no reason to stay with the Lannisters if they’re not going be true to their word about paying him.
So then what does that leave? Where should Bronn go?
One fan’s theory seems to think that he should at least attempt to ally himself with Daenerys and her forces. But there are a few issues with that. First and foremost, he shot Drogon in the shoulder, with Dany atop his back. That surely won’t win over the Mother of Dragons. Second, he was seen fighting for the Lannisters before the attack on the supply train, so it’s well known to Tyrion and many others where Bronn’s allegiances lay. Third, even if he’s quit of the Lannisters there’s nothing to say that he wouldn’t turn on Daenerys and Tyrion the moment it seemed like they were on the losing side. He’s a survivor, remember? Bronn will go on his merry way if he feels the need and support anyone that’s going to pay him, no more and no less.
It would be interesting to see Bronn join Daenerys’s forces, though you have to reason that he’d be closely watched and kept under close guard at all times. Even if he did prove himself he’d have a long to go towards making an apology for the wound he caused to Drogon. Something tells me the dragon isn’t going to be so quick to forgive the mercenary.