Is it permissible to boycott the paternal cousin who divorced one's sister and has children with her, given that he is also the brother of one's brother's wife, and would boycotting him be considered a sin, especially since his family is causing harm?

It is not permissible for a Muslim to forsake his brother for more than three nights. This is particularly emphasized concerning relatives, for abandonment prevents the forgiveness of sins. Do not abandon this man simply because he divorced your sister; mere disgust is not something one is held accountable for.