Sorry to hear about your bother with the neighbour. I have 8 cockerels of various breeds around the place, and some of the young silkies are turning into boys too. Bearing in mind we were there before two of the neighbours, because they converted farm barns, there is also two houses next to the field. We live down the farm track!!!
No one has ever said anything, or reported me, although I was ready for it happening. Except a visitor, who one morning enquired as to how many roosters I had. My reply .....3. Well, he was a townie, and would not know a cockerel from a hen.
Are they in a field, or your garden? I would say if its a smallholding and a field, there is not much anyone can do surely? A residential area, probably yes, if there are enough complaints.
When you say the neighbour is trying to get you to move - why exactly??