I've considered EA playing a role, and to me it makes sense only in BDI theory.
I believe Burke was probably sexually abused, but I won't speculate by whom, I have no idea. Anyway, as you stated, EA is for the gratification of the one being asphyxiated. So I've considered what if Burke's abuser had taught Burke to choke him. The reason being, if the abuser, during the abuse, had one hand on himself and the other hand on Burke, how's he going to choke himself (assuming that's his thing)? So just teach Burke to do it, right?
Simply use the length of cord, without the brush handle so there is no freaky sex toy to be found by anyone who may stumble across it while doing any number of activities in their little play room, only an uninteresting piece of cord. How would he teach him? I assume just tell him, "here, wrap this around my neck like this and keep pulling it tighter until I nod at you"..."yeah, people like this, it's fun"...
Christmas night when Burke is playing doctor or whatever he may have been doing with JBR after coaxing her out of bed by promising to get her favorite snack for her, which he does, he says "hey, try this, people like it." He doesn't tell her to nod and she doesn't know to. He pulls too tight, explaining the first marks on her neck, and limits the blood flow to her brain, causing her to pass out. Once he realizes she's passed out, and won't answer him when he talks to her, he shakes her. She doesn't come to. He gets a little scared. He sees the hammer, decides that will wake her up. He swings the hammer entirely too hard, causing the head trauma. She goes into a seizure from the head trauma, convulsing and puking. The 9-year-old tears out of there, running back up to his room in an attempt to pretend to be asleep. Mom, still awake and packing, hears him and decides to investigate. These kids need to be sleeping, not up playing with their toys, we have an early flight tomorrow. She goes to the kids rooms. "Burke, where's your sister?" He pretends to sleep, not answering her. "Young man, I know you're awake, now tell me where JonBenet is." He rolls over, trying to appear sleepy-eyed. "I dunno, I think she went downstairs." Patsy looks at him, skeptically "You think, or you know?" He realizes he'd better tell, but doesn't want to get into trouble. "We went downstairs and we were playing in the basement and she started tying a rope around her neck and I told her to stop but she wouldn't, then she got sick and puked." "Where is she now Burke?" "Still down there". Patsy hurries down the stairs, Burke following. The scream reported by the neighbor is made by Patsy when she sees the little girl's body.
Sorry, I can get long-winded...anyway, that's pretty much my line of thinking at this point