I guess your neighbors are very very lucky not to have me as a neighbor. With what they did it would have only made me dig my heels in deeper and stand my ground.
I have always been a live and let live kind of person. One of the few in this world I suppose that does not care much what goes on on the other side of the fence, so long as it is law abiding.
I have a dog hater as a new neighbor. She should have looked through the chain link fences and seen all the dogs in the area BEFORE buying the house. Mine barked at a fox at about 5:45 am one time and she called the cops. The next weekend I tore down a 2 year old chain link fence and put up a privacy fence. Only after I told her to get bent.
Never could stand folks telling other folks how to live.
I am sorry you have to give up the chickens.
LOL my cranky neighbor must just be beside herself every time one of my hens lays an egg.
Hugs for your kiddos and perhaps a bunny would not be a bad idea.
edited to add...
oops missed the part about the allergies. No bunny then.