Like a pp, I am also wondering if Jaimie knew Bran was still alive? He has been back at Winterfell for some amount of time so someone could have sent news to Kings Landing but not sure we ever saw that happening?
I reckon Arya doesn't want to admit to Jon - her favourite brother - everything she has done. At least not yet. Hence the downplaying of the killing. It's like she had a chance to go back and be the little girl again for a few moments.
Brown selling out Tyrion and Jaimie and trying to kill them would fit the spirit of the books I think. But I suspect in the TV show it may be a plot device to get him north as that is where most of the long-standing characters have now congregated but he won't go through with it.
So many complex issues to tease out with Jaimie's arrival in an area where everyone hates him (and only Brienne and presumably Bran have any idea of the more honourable motivations behind some of his actions) and the implications/possible reactions to Jon's parentage (if they manage to persuade people it is true on the word of a vision and an old diary...). I worry slightly that due to the short time left in the series some of these are go to have to be resolved quickly to move the plot along rather than properly explored.