I'm reading a book that got me thinking about the difference between なぜ, なんで and どうして; as context, the speaking character (ガラザイア) is asking another character (ギルロア) why he killed ルイーゼ様. The second character was close to ルイーゼ様 and her brother (レメディオス), and all the rebellion trusted him; I don't think it's relevant, but while the speaking character is convinced the other killed ルイーゼ様, it's never said so, and it could well be he didn't.
The narrator is a third character, which is secretely listening to the conversation. This is the exchange that got me thinking:
Not sure if it's relevant, the speaking character is growing increasily frustrated and angry.
I found this answer explaining that while なぜ just asks for a cause, どうして and なんで also asks for method; but in this case I really feel like the method is completely irrelevant, both because she already said it in the first quoted sentence, and because really it doesn't seem to matter how he betrayed her, but just the fact he did.
So, I was wondering what's the meaning and/or nuances of asking both なんで and どうして (and, at the beginning, なぜ) in this case.