Ambessa is so so abusive to Mel precisely because she thinks that this is what good parenting is. She is embodying hyperrealization of cultural standard that is fundamentally messed up and built on violence and conquest. They are the parts of herself that bring her alive, and that she finds to come naturally to her. How do you raise a daughter so deeply unlike you, when your way is the only way you know to stay alive, especially in Noxus?
Because Ambessa loves her daughter! Deeply! Stupidly! And that love does not point her to kindness or adoration the way so many of Arcane's other parents are pointed - it points her to fear and violence, because to Ambessa, that is what love is. Her voice lines in the game literally indicate this - she considers motherhood and battle to be two sides of the same thing, conflated into one cohesive whole. How can her mothering be anything but a fight against their nature, to ensure they will survive anything she fears?
And the Black Rose is then the embodiment of those fears made flesh. There is actually something after her, her family, her children. There is something that has killed her son. There is something that has taken her daughter. What kind of mother would she be, if she did not fight for them?
Nevermind that this was an enemy she delivered to their doorstep. Nevermind the ways her own politicking have places Mel under risk. Nevermind that Ambessa has herself triggered actions that placed her daughter directly under a gun. This is the way to ensure her family will be safe.