The Bettesworth (and variants) were often found in Trotton, Midhurst, Woolbeding, Chithurst, Iping, Elstead, Linchmere, Treyford, Kirdford, Fyning, Rogate, Turwyke, Harting, Easeborne, and Wollavington in West Sussex, so it might help you to look in those directions.