In that part of his statement that I heard, he called her Patricia several times...always in context of his efforts to be in communication with her and get his points across to her that he was sorry about what happened to Jon Benet, wanting to do so before she died, wanting to be assured that she either read or was aware of his writings.
It startled me when I heard it. It sounded unlike what I'm used to hearing...Patsy is Patsy to the world, though of course her birthname is Patricia.
It fits with what you said he was trying to do...place himself inside their world. At first, he was gloating over having gotten the best of them. Most recently, it's like he was trying to participate in the grief in their world...whether grief all these years over Jon Benet, or grief over the recent passing of Patsy.
If he knew Patsy was dying, and he most likely did, that would explain his timing for trying to contact her. He wanted her to know. There wasn't much time left to continue the game, and he wanted to be a major part of the end of Patsy's life. I'm glad she never knew who he was. He's sitting back gloating because he thinks he's corresponding with Patsy. He thinks he has that last bit of control at the end of her life. It's great that it was a scam on him by the police. The only thing she knew was that there was a suspect that wanted to meet her. That didn't happen thank goodness, and now he knows that Patsy didn't even know his name.