If the dog has shown aggression towards your child in your presence, you can't trust the dog to be around your child. And when it's a child's safety or a dog, there's no question - the dog needs to be re-homed to where its instincts won't harm a child or chickens. I know a lot of people think little dogs are cute, but dauschunds are pint-sized hunters. I'd never have one with kids. My best friend's daughter was bit on the face by a dauschund - it clamped its jaws on her cheek and despite everything they tried, it would not let go for more than five minutes (until the EMT and police got there). More than $50,000 in reconstructive and plastic surgeries later, she still has a scar that will require more surgeries, plus therapy to get over the trauma. She went from being a sunny, outgoing 3-year old to a withdrawn child that is terrified of any animal of any size. The thing that infuriates me is that that dog had bit children before and the owners hadn't done anything to prevent further attacks.
I'm sorry for everyone involved - the silkie, the rabbit, you, AND the dog - it's a tough situation to be in. I hope you can find a solution that will work for the best for everyone.