It's obvious that Lord Tywin knew Arya was a Northern girl of noble birth. Then would it have been really far-fetched to assume this was another Northern noblewoman they were searching for. I guess making the assumption that Arya ran from King's Landing all the way to Harrenhal may be a bit far-fetched. But, he still should've tried to find out more about her. What if she was related to one of Robb's vassals, who they could hold as a bargaining chip?