Little Finger wanted to use Brienne in one of his plots, he sees some angle can be exploited.
Sansa knows that although she doesn't know what Little Finger plots, Little Finger would not have spoken on getting Brienne involved unless it was for his own gain, and likely everyone else's loss. Especially because in her capacity Brienne could act as another advisors to the sisters, which could interfere with Little Finger's plots. All Sansa knows is that if Little Finger talks of Brienne, he intends to exploit her somehow so to protect her, Sansa sends her away (wherever that being is irrelevant). Really this was the most obvious bit in the last episode it came right after Little Finger mentioned Brienne, it should have been a red flag, it should have also been apparent that when Arya met Sansa the reception was warm, and the 180 degree turn in behavior was stifling and not natural, the most obvious piece of information was where Arya got that letter from. There's no way when Arya brought it to Sansa, that Sansa wouldn't have concluded where it came from, and there's no reason that Little Finger would have that note that was not nefarious in nature, in any way shape or form. It never made sense that Arya would actually go after Sansa, and it never made sense that either of the sisters couldn't rationalize the reality of the events as they unfold. Plus Bran can see everything, so there's that.....
