His father is dying and you're some asshole that's fucking his sister who's trying to intrude on the situation.

Clearly I don't actually believe that, but it's helpful to start understanding the dynamic by internalizing the other guy's point of view.

In this situation the right approach is to walk away, you can't and shouldn't try to get between someone and their father in a moment of crisis. His sister has standing, you don't. She has to step up and deal with it, or not, at her discretion.