Is it permissible for a divorced woman with a daughter to marry secretly, without documentation "by means of a written marriage contract with a Mutawwi'", to avoid losing custody of her daughter?

If the former husband is a transgressor (such as drinking, stealing, and committing adultery), then he has no right to custody, because a transgressor cannot be entrusted with the care of the child. Custody is established for someone whose condition is concealed (i.e., of good moral standing). If there is no other rightful custodian, and the mother fears that the transgressing father might seize custody by force or by law after her marriage, then there is no blame upon her for concealing her marriage from him, because declaring the marriage is not a condition for its validity according to the majority of scholars; rather, it is valid if its pillars are met. However, if there is another rightful custodian other than the aforementioned father, then it is not permissible to use deception to circumvent his right.