Im so sorry that your neighbor is so damn dumb. I love roosters more than hens and get angry when people complain about roosters but not constantly barking dogs or something. My uncle used to live on a farm and had a new neighbor and got angry when they saw barn animals even though the house specifically said it was next to a farm. The neighbor is dumb, and so is yours. City people are dumb when it comes to chickens and especially roosters, this is bad advice but if I were you I would educate her on roosters then ask her why she even moved into a farming community, people who own chickens generally live in a farming community, its like getting mad when you smell poultry on a poultry farm. I would consider having a large coop on the farthest part of your property, you might want to consider placing thick old towels and blankets on the parts of the coop that face the neighbors direction next, this is to dampen the sound. Because it sounds like you need an acre to hold 1 roo but you only have .5 acres, I would do this as soon as possible. If she tells the town about you holding a rooster when your not allowed to then your in even more trouble and it wont take long to get ugly, although you might get away with it by doing things to dampen his crowing. Hope that helps.
If a large coop wont do anything, this might sound even weirder but you might want to consider asking her if planting sound-barrier trees/bushes like Oleander on your side of the property to help with sound dampening, both are easy to grow. I hope I could offer help.
One thing you also want to do is be nice and respectful, for example lets say she is annoyed at the crowing sound and wants to know if you could move the rooster farther away on your property. Its good to respond with "Thanks for telling me, Ill do that so its not an inconvenience." Instead of saying "I wont get rid of it." Its frustrating when city folks move into farming communities because they don't understand the law of the land just yet and how they should not be upset because of livestock, then why they moved into a farming community then get mad at smelling, hearing or seeing livestock.
Also remember how she never asked you to get rid of him, she still cares that you care about him and while that's still going on, showing that you respect her concerns might make her stop complaining hell maybe you 2 will even be friends and you might even get to keep the roo in the end.