Pretty sure that was Arya who sent Brienne off. Sansa just talked with LF about how keeping Brienne close would be the smart thing to do, and right after Arya said pointedly that she can be anyone, even Sansa. She needed Brienne gone to get close to Sansa, I guess. To what end, I have no fucking clue.
I don't think Arya has killed Sansa and taken her face.
Littlefinger told Sansa that as long as Brienne was there, she would be honor bound to protect them from each other. To me, Sansa sending Brienne away was about removing that obstacle so she could potentially deal with Arya.