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sweet sparrow sorry to hear dd1 is hurting herself. Does she a have a stuffed animal to snuggle with and act as a distraction. I wouldn't do as you recommended because that could open a whole new set of problems if people read or hear about it...if you get my drift.
I do feel it's perhaps for attention or for her a copping mechanism, I would talk to your doctor if she continues to do this and I can only imagine how concerning it is.
Can she look at a book before falling asleep/napping to act as a distraction.
Have you asked her why she does it, I know we sometimes look for solutions but perhaps she will answer why, she seems very intelligent from what you've said.
Does she want to be more involved with helping with the baby or perhaps resents the baby?
Good luck