Hi, and welcome to Using English.
If you know her well enough to know her sexual orientation and personal foibles, you probably know her well enough to write "Dear Jane:"
Her expectations are unrealistic. Salutations are part of a business letter. She can choose whether she wants to be Ms. or Miss, but to insist there be nothing is an unfair requirement. (I would toss the "to whom it may concern" ones out too; if you know my name, use it.) Does she refuse to write "Dear Mr. Smith" on her outgoing correspondence?
I had to work with someone who thought that using a lower-case first letter for her name showed that she was being humble. In fact, every time we typed her name, we had to go back and over-ride the auto-correct feature. Every time we had her name in print, we had to make the effort to tell the editor "she spells is that way." We knew that others reading it would assume we'd made a typographical error, and since few people can resist the urge to tell you about your mistakes, we had to tell all of them that too. So, in the end, her "humbleness" required a great deal more work to accommodate than if she'd used a capital letter like everyone else. In my opinion, people with these issues, like your friend here, have ego issues that they address by requiring the rest of society to jump through hoops to come up with alternatives that suit them.