I hated Brienne crying too. Perhaps a better written conversation would have been after he wakes her up instead of just leaving her.
Jaime: I’ve done a lot of awful things for Cersei. I don’t know if I’m a good man, but I’m still trying to be a better man so I must try to stop Cersei. It’s my duty as her twin brother to stop this. I’m the only person she may listen to.
Brienne: I understand. You wouldn’t be the man I thought you were if you didn’t try to prevent it. I would come with you, but you know I’ve sworn to protect Sansa. (No crying)
Such a huge difference.