You think the rats ran out and those cats just decided to stop hunting? You really think the area is entirely devoid of all other predators as well?
I am going to try to have enough self control to not try to talk any sense into you any more. You clearly don't have any desire to be reasonable...
A loose bantam chick is at risk from many, many more predators than a dog or a person. A housecat, a raptor, a weasel, a corvid, a snake, or many other things would find it an attractive meal.
The cutting human hair analogy doesn't really fly with me, either. Human hair does indeed grow back...
Keep in mind you currently have zero evidence that this neighbor harmed your chicken. All you have evidence is that your chickens have done harm to your neighbors property (even if you don't think it was much).
Free range chickens are often happy. They are also often taken by predators. And...