Well, Tyrion has a duty in King's Landing, but he can still visit Casterly Rock if his wife is there in Jaime's custody, right? (Assuming Jaime's reading lessons with Tywin took place there when Tywin was Aerys's Hand. Seems like the Lannisters grew up there instead of at court.) Or they could just find another Master of Coin, Tyrion basically got the job to fill an opening and give him something to do. The point is, Jaime's not even trying to sound out the options, but he does feel confident to speak his mind to Tywin for his own interests. Agreeing to help Cat at swordpoint doesn't mean anything, but he chose to affirm the vow to Brienne before leaving Harrenhal. He's gotten back to King's Landing, almost intact, the ball's in his court now. I can only think that's why Brienne hasn't introduced herself to Sansa, she wants to know she can help her and she can't exactly insert herself into Sansa's Lannister household. And taking a page from Sandor's book and just snatching her up to go wandering isn't the best idea either.