To me the biggest gut-level warning sign that he is lying about the Ramsey case is his breathless claim to have "loved JonBenet". That implies an obsession with her public image like a deranged fan, not what I would expect to hear from a calculating child rapist and murderer who has for ten years evaded capture for the most famous child murder since Lindbergh's baby. This guy was not a drifting transient - place him in Colorado in December 1996, which should be easy to do - and then I'll take it more seriously.
I've said this before, John Mark Karr with the Thai police, cameras and microphones surrounding him reminded me of Gloria Swanson in "Sunset Boulevard": "Mr. Demille, I'm ready for my close-up now". Only, I don't know if the confesion is for real. He just seems to be obsessed and living in a fantasy world.