No she cannot petition her. Her name is not on the birth certificate. Your mother did not raise her and provide care for her all her life, and she is over 18.
A DNA test doesn’t matter she is not listed as the legal parent. You are probably misinformed on your sisters legal custody is with the cousin. Regardless of it was an adoption in court or not. For immigration purposes your mother has prove she provided care for them while she lived in the USA.