I dig Jane because there's also another tie there.

John and Jane Doe, typically used when there's no personal identification.

Though I also feel like we're losing something because of the medium. Is it "Gene" like genie or "Jawn" like Jeanne?

I live in the rural South, it could genuinely go either way in my head, though I lean towards "Jawn" for the better flow from "John".

EDIT: Okay, I cannot decide which makes more sense as a pronunciation guide, if you've played Bayonetta, you've heard it pronounced.