I just read the ransom note again, with the idea that Karr might have written it.
It doesn't add up. Karr thinks he is in love with Jonbenet, and that's what he calls her - Jonbenet. It the note, she is never named, she's just "your daughter".
Karr thinks of Jonbenet romantically, and yet the note talks about beheading her, and "she dies, she dies, she dies". That doesn't add up to Karr's way of thinking, either.
There's nothing in that note that shows the writer as having fantasies about Jonbenet. The writer distances himself from Jonbenet as a person, and focuses on John Ramsey as a person.
Also, it's hard to imagine geeky Karr referring to himself as a "fat cat".