You need to be much much firmer with her OP, I`d give the dsi to a charity shop next time.
She really will end up pushing that poor dog too far.
Think about stair gates for your doorways, Ive got them to separate my soon to be born baby and my huge slobbering hound.
Can you sit her down and talk to her when she isnt being mean to the dog, in other words when shes behaving not having a strop? So you`re both calm and not in the middle of a row.
She really must not be allowed to hurt the dog again, I couldnt bare the thought of anyone hurting my big smelly slobbery boy.
I hope you dont feel like everyone is picking on you OP, obviously theres lots of us animal lovers around tonight, but it does sound like your daughter is a bit spoilt, (I was an only child too, I know what its like to be treated like a perfect princess) and likes to be the centre of attention etc, it needs nipped in the bud, cos it seems clear that she could quite easily grow into a very unpleasant adult.
If you do decide to re-home the dog, then I would also re-home her other pets too, she`ll learn that her actions have consequences.
I hope you get it resolved so that the dog gets to stay and live in peace, and your daughter learns to be kind to it, although sadly its probably just waiting for its next beating and on edge when your daughter is around.
Again, I hope I dont come across like Im having a go, but something needs to be done