Reading back is refreshing my memories of this case. May I toss in a theory?
The post a year ago about refrangent material found in JB's vagina leads me to this: cellulose may be refrangent. Many plastics definitely are. The missing/replaced doll was plastic. The duct tape may have been taken from the doll; the brown and blue fibres found on JB match the colour of the doll's clothing. OK? So we have
BR smacks JB with something, too hard, knocks her out.
Parents (or likely just Patsy) find out. What to do? what to do?
* I agree the garrotting is to cover the assumed broken neck. The cord used was the original hanger cord of the painting that was removed; it may be many years old and you'd not expect to find any more of it in the house. Not in drawers; maybe on other pictures but they never looked at that did they?
* Nobody just murders a child, right? it is always rapists, right? so she needs to look assaulted for a cover-up to work.
* THE DOLL. She's prepared to strangle her child, believing her already dead, but she's not prepared to digitally rape her. So she uses an arm or a leg of the doll.
* explains refrangent material
* explains bruising and bleeding
* explains blue fibre matching the doll's clothing in the vaginal area
* explains why the doll has to be removed (and later replaced with a new one)
* explains vaginal opening being wider than expected from a finger, which is what suspicions of previous abuse are based on -- it is a 15" doll
Then, being a mother who in her own strange way does love her child, she arranges her clothes neatly.(Oversize pants? needed to cover the Pampers so nobody suspects; no big mystery to me).