Well, technically it isn't a tradition of the pre-Schism West, as Sarum Cathedral was dedicated in 1092, and the Sarum as known from the texts dates from New Salisbury in the 13th c. That it is essentially no different than Pre-Schism Frankish and Celtic-Saxon Roman traditions is witnessed to by contemporaries, but the Use itself is definitely post-Schism. All surviving documents of the Sarum use are post-LePoore, in fact.