So sorry to hear about your situation. I am definitely NOT taking your neighbor's side, but is his complaint stemming from a certain aspect of your having chickens, or just complaining? If it's the former (noise? smell?), is there anything you can do about it? Not to appease him, but to show that you have done your part - and more! - to be a good neighbor. If it's the latter, maybe he just likes being a PITA, and unfortunately, there are people like that. (I have relatives like that, so I have first hand experience.)
I think your ideas sound like a good solution to a bad problem. My worries about the umbrellas would be concerning wind. Get the most sturdy ones you can find. Maybe if strong winds are forecast, be prepared to take them down if you have to. And, thank you for taking such good care of your chickens.
Again, so sorry you have this problem! You've made me realize how lucky I am to have the neighbors I do.
And, 3KillerBs, I agree with your comment entirely!