In my experience of neighbours with pets, they usually put the hutch or coop/ house at the end of their garden, which of course is very near your house!
Maybe they don't hear the racket the new pet makes all night long, and maybe they need to know their new little pet is crying all night.
Maybe the foxes are cicrcling, and frightening the life out of it. Maybe it's cold, and lonely, and super sad.
On no account put poison into their garden..... Whoever suggested that must be criminally insane..... I mean seriously, that's so off the wall, it's boggling... What if another animal or bird ate it, or their child or someone else!!
Or it got washed down into the ground water..... Please don't listen to that loon.
Fwiw, I know it's easy to get attached to pets, and see them as part of the family (having pets in our family ourselves!) but they really aren't the same as having a toddler. Let's keep it real here.
OP I hope you get some resolution soon, being sleep deprived with your own kids is bad enough, but from a neighbour's pet is crazy making!
I'd have a word.