Is she getting any money /benefits from the state with having the child?
Is that one reason she drags kid around and has not dumped somewhere with a family member?
Maybe offer to take the kid if she signs off parental rights?

What about drug testing her if she is going to be in your home.. her actions are more serious than just a liar...
And she could also be "self" medicating /acting out due to mental/emotional health - it can be hereditary..
Require a full checkup and include mental health, it might be tough to make it happen, but could save her, the kid and your family sanity..