This is a bad question.
The normal answer would be "No, sorry" just as you say, or "I don't have any."
I can't imagine ever answering "none."
The following is a test for English class students to choose a correct answer:
Q----I want some more paper. John, do you have any in your desk?
A----__________(Nothing/None).
The students have to choose between 'nothing' and 'none'. I selected 'nothing', while the given answer is 'none'.
I know someone could say: 'No, sorry'. But it is not there for us to choose.
Is 'none' the right one there? Thank you.
This is a bad question.
The normal answer would be "No, sorry" just as you say, or "I don't have any."
I can't imagine ever answering "none."
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
If those are the only two choices, then it would have to be "None", being an abbreviation of "No, I have none". Having said that, we would rarely say that. We would say "No, I don't have any".
I agree it's a bad question, or rather, it's a bad choice of answers.