Then by the wording of the rules there I would assume you can't have chickens. Best to call the county and not bring it up to your neighbors til you figure that out.
Do I understand right that your neighbors rules do not apply to your zoned property or are you part of the covenant that includes your neighbors property.
If you know where you stand legally and your neighbor isn't onboard just throw eggs at him.....not literally of course. Give him a dozen eggs once they are in full production, might take a little while to get to that point but a gift is a gift. 4 hens aren't to messy and there are options for...