Her primary concern is the health & safety of her child. She cannot (and should not) be culpable for the actions of a deranged adult, and therefore her responsibiities should be towards protecting her child.
I knw it is easy to say and she probably feels stuck between a rock and a hard place, but he is clearly using emotional blackmail to get attention and his own way. This is affecting the child now and will continue to do so for as long as she lets his manipulative behaviour carry on in front of the child.
It sounds crass, but she really needs to set him an ultimatum, or at least some targets and clearly explain her reasons for doing so - along the lines of the childs mental development and induced stress. He clearly needs to get some professional help and he just needs a push, with the reassurance that once he gets help and some stability that the child would love to see more of him and she would be happy for that. Providing there are signs of improvement.
It is difficult I know, but really she has to look out for the child and let the partner make his own adult decisions. Even if that little push makes him do something stupid, she should not place such a loathsome burden upon herself or her child as she and the child will all suffer as well.
Crunch time.