Do you have someone who could take on your 2 extra chickens until things simmer down? I'm sure your neighbors just complained in anticipation of having something to moan about later. If they said things were fine now, you might hold them to their word - and they didn't want that.
I think once you put the run cover on and use stall dri etc., you would pass muster anywhere. I find it hard to get upset over maybe one hen annoucing her egg. Even if they all did it, it's not like it would go on all day. They only lay one egg a day at most. There are always people who's only purpose in life is to ruffle their feathers and be atch . Unfortunately, we all seem to live next to one of them.
Do they cause any aggravation to you (loud music, pets etc.)? If so I would kinda blow off their next complaint and respond with " Your dog seems to find alot of bark about in the middle of the night." or " How often do you have those loud parties on your deck," kinda thing. Make them go on the defensive. That usually defuses them at least for while.
Good luck to you. There are people on this site would you dearly love to be able to legally have just 4 hens.