I never bought totally that idea she was only a human woman. Humans don't have maroon/reddish eyes. I suspect there were more "things" planned to this character, but Capcom, maybe for laziness, maybe for lacking of interest, lt them fell apart.
About becoming a demon: I guess we make more of a fuss about that than her would do. She is alone, she's not tied to anyone and basically defenseless and she just can really count on herself. Maybe she would become disturbed about that change, but if she could deal with the idea she has the blood of the man she most hated running in her veins, being a demon wouldn't be worse.
I know people says is a kind of cool she is a human and they refuse let Capcom change her nature,But it's Lady's humanity that most plays against her since since fans rejects her and doing anything with her because she's human. Being human in DMC is a bad thing.A radical but tasty change wouldn't hurt the character.