Hi Donna, I'm so sorry this happened to Ian, I am so sorry for your loss, and my heart goes out to you and your girls.
I agree it's sad that the dog has to be put down, but he is a danger to your girls and Sebastien, and if this didn't happen then you would never be able to trust your girls or Sebastien around the dog, which would mean you wouldn't be able to use your own backyard, which isn't fair, everyone should be safe in their own backyard. Did you have problems with the dog before this? What is your neighbours like, are they reasonable or are they the type of people who would retaliate?
I hope they are reasonable, and realize that their dog is vicious and can't be trusted. I hope they wouldn't do anything to Sebastien. They are clearly in the wrong, and should have had more control of their dog, and made sure he was safely contained either in the house or in the backyard. I hope you're girls are okay when you tell them.