Is it permissible to register a child of unknown parentage—whose mother was raped by Qaddafi's brigades—in the family register after sponsoring him, knowing that he was suckled by the wife's sister?

This child is attributed to his mother, and she is not permitted to relinquish his lineage. However, she is permitted to relinquish his guardianship to you, and this guardianship is an act of worship for which you will be rewarded. It is not permissible for him to be attributed to you, as this is the forbidden adoption that Islam nullified. Breastfeeding establishes mahramiyyah (prohibition of marriage) but does not validate adoption. If embarrassment is feared from attributing the child to his mother, he can be attributed generally as "So-and-so bin Abdullah," while his original lineage to his mother remains preserved.