Is the questioner considered a sinner in the eyes of Allah if his Christian wife and their daughter leave, and the daughter becomes one of the People of the Book like her mother, even though he strives to maintain her Islamic upbringing and raise her upon it?

A marriage contract with a Kitabiyyah (woman of the Book) is not valid without a guardian (wali). Therefore, the questioner's marriage, being without a guardian, is invalid. He must renew the contract in the presence of her guardian. His previous marriage is considered a شبهة (doubtful or شبهة-based union).

Marrying Kitabiyyahs entails risks to the children's religious upbringing. As for attributing the girl to her father through fornication (zina), this is incorrect according to the majority of scholars. However, Ishaq ibn Rahawayh, Urwah, Sulayman ibn Yasar, and Abu Hanifa permitted it, such that the child is attributed to the fornicator. According to this view, the girl is his daughter, and he is responsible for her. According to the majority of scholars, there is no relationship between him and her, and he is not responsible for her.

He must repent from fornication and stay away from this woman unless a new, valid contract is concluded with her, with the permission of her guardian.