Fair point, but again, her history with Vergil is on the lie Arkham told her, nothing more. There's nothing else beyond that, and she has accused him (ever so passive-aggressively) of being tied to her mother's death. That is all she cared mostly about.

I already explained that Lady doesn't care what his lineage is, it's pretty much obvious she doesn't; if that were the case, she'd be kicking Dante to the curb, too.