Many proper and common nouns in English could be used as a verb. One simple example is
A: What can I do for you, sir?
B: Don't sir me. I don't like being called "sir". It makes me feel old!
The sentence in your example could be rephrased to:
Don't you (dare) call my first name (as if you were my close friend), backstabbing little traitors.
First names are usually used among close friends in casual settings. Scarlet doesn't want Stuart to consider her as a friend (as she considers him as a traitor). That's what "Don't Scarlet me" basically means.
The question and answers posted on ELU, “To science the sh*t out of something” will be helpful.