I cannot find anything about her.
From the Christie's Web site.
Lady Josephine rests wearing a white dress with criss-cross bodice. The signs of status are her pendant and the white of her dress.
Keywords: Beechey, Josephine Sophia Newton, Newton family, Cooper family, British, high straight coiffure, sweetheart décolletage, long upper puffed sleeves, cuffs, Empire waistline, close skirt, necklace, wrap