Yes, Parents of extremely obese children should lose custody!
Therapy and education programs should be a first line of offense in treating obese children, but when a parent repeatedly ignores advice and guidance for the child's well-being, government intervention may be an appropriate next step. State intervention ideally will support not just the child but the whole family, with the goal of reuniting child and family as soon as possible. That may require instruction on parenting, despite the discomfort posed by state intervention, it may sometimes be necessary to protect a child.