I think Arya knows that Littlfinger set the letter up for her, so she also knows he could be spying on her while talking to Sansa. So all of the Winterfell squabbles are not real, it is just a play for Littlefinger.
When giving the dagger, Arya uses the opportunity to send a message to Sansa: "Now when you have seen what I can do, let's team up. Perhaps I don't have to use your face after all, since you can be stronger yourself. Just show me a target to use the dagger on."¨
That's how I would try to read it, otherwise the Winterfell scenes don't make much sense to me.