Lewis never intended the Chronicles of Narnia to be an allegory of the Bible or of Christian life. I doubt that he intended a direct connection between Edmund's temptation by the White Witch and Eve's temptation.
A more direct link between the garden of Eden and Narnia is found in The Magician's Nephew, and even there we do not find one-to-one correlations between characters and events. There it is the Witch that eats the stolen apple, but she is not a daughter of Eve either. There is a suggestion that she is Adam's first wife Lilith, who appears in some Jewish legends, which opens up many possible interpretations of what Lewis was trying to say.
