Seriously, what part of ANYTHING he knows about Arya would make Gendry think she'd want to be a lady. She's spent her ENTIRE LIFE actively trying to avoid that exact thing.
Also, how fucking could you Jamie?
That sure isn't what he said, though I do think he THINKS he going to be able to defuse the situation, and I do suspect he will end up killing her, he still broke Brienne's heart.
I want to believe Jaime lied to Brienne to keep her from following him. I want to believe he aims to save King's Landing, just like he did all those years ago when he had to kill the Mad King to do it. I think he doesn't expect to come out alive this time...
Also, how fucking could you Jamie?