|You know who's fault it is that dd was hurt?
The supervising adult.
ALL dogs have a point at which they will snap or bite. It is foolish to think otherwise or take a dog for granted. Having a dog and small children in the same house is a huge responsibility. I do NOT allow my dd to threaten or harrass my dog, period. Constant vigilance is a drag, but I love my family and its worth it. If dd is feeling especially rowdy or if I can't pay close attention, the dog spends a little time outisde or on the other side of a baby gate.
I do think the OP should consider rehoming the dog unless her family is willing to change the way they handle the dog and child. I have a close friend who adopted a lab, and let her toddler get too rough. She walked in one day to find the lab with the girl's head in his mouth. Terrifying, but the babe was unhurt. My friend did some soul-searching and talked with professionals, and instead of kicking the dog out they changed the house rules. The dog is a terrific dog who was put in a bad situation by the adults. A dog who isn't being protected has to decide when to protect himself, and that is a failure of leadership on the owner's part.
Training and supervision. Training and supervision. Training and supervision.
Repeat x 100.