What are the correct Islamic legal procedures for sponsoring a child born as a result of an assault, given the refusal to register the child under the sponsor's name and the request for a chromosomal analysis of the child?

Sponsoring and caring for this child is a great act of devotion and a form of kindness that God loves. You have done well by avoiding the forbidden act of adoption. The child is attributed to his mother and inherits from her. There is no harm in a general attribution such as "so-and-so son of Abdullah" while preserving his original lineage to his mother. As for the analysis of aggressive genes, it should not be an impediment to his care, and you should not pay attention to such whispers [baseless thoughts]. Be aware that such assaults might be a result of leniency in dress or being alone with the child.