1st of all, I am so sorry you have such an awful neighbor. That jackass has complained once, now that door is open & there will be more complaints unfortunately. I know, I lived near jackasses once myself. For you to get approval, then later be told to change what was previously approved, to please some jackass, is not a good sign of things to come, but it sounds like you already know that.
My pens have 1/2" x 1/2" wire mesh, hardware cloth, keeps mice out as well as predators.
I have bought Pet Gazebos for my chickens to enjoy shade, and they're movable things. I even bought shade cloth and wrapped around the sides to keep dry inside during winter but those photos are gone. They love taking their dust baths in there, I position the perch towards to front so they don't dust bathe in the poo poo. These gazebos can easily be moved for grazing, but mostly I have them set up inside their fox & hawk proof pens.
I have considered a type of fabric, I think it may be called Sunbrella, it is like your umbrella, that I can make a rolling retractable "awning" but I haven't made it yet. Since that will be removable & my Gazebos are movable, they're not permanent structures.
So anyway...my question, is there a particular reason, like a possible noise, compost heap or mess, odor or Rooster crow factor, or is this just one of those jackasses home all day looking out windows for something to bitch about? IF it's just a miserable person, nothing you can do will work. But IF there is a reason, it may be possible to reach some type of "live & let live compromise" but you will have to make effort to initiate "polite with no anger" conversation, and that may be impossible at this point. I don't know your particular jackass, I don't know if offering fresh eggs or awesome composted garden gold soil for jackass's garden would help or not.
In the mean time, is there a way to revise your pen set up? How big is your yard & coop? Is there a way to actually shift things, so that the only thing that jackass could see, would be the back nicely painted wall of a coop...and not be able to see the pen or flock at all? At my previous home, I had 6 foot stockade fence around the backyard, but I also arranged 3 large sheds like a courtyard, with their pens in front. So envision 3 sheds creating like a big U shape, openings with pens facing the middle of my backyard. If any neighbor on 3 sides did look over the fence, all they saw was the nicely painted back of a shed. I planted shrubs along the sides of sheds and partially down sides of pens, not right up against pens but close enough, so even if anyone jumped up on my fence, they still could not see what birds I had, which back then, were only Homing Pigeons, but in a neighborhood I didn't want any complaints. If you have a Rooster, obviously, that's a whole other issue lol.
Is your coop elevated or built affixed to a foundation? I've moved a few sheds using long pvc pipe, chain, the truck, a come along & few strong guys, in a couple hours. If moving or repositioning the coop is not possible, how about changing what side the pen is on? I know it means some construction remodel, but if I could build a new pen & new opening from coop into pen, out of that neighbor's view, that's what I'd do. I had a busybody jackass neighbor once, old in ill health, all she did was look out the window and find things to bitch about. I get it...old, in pain, everyone you love is dead, so when you see kids playing, young folk walking dogs or neighbor across street gardening, doing things you can't do anymore, it pisses you off, so you bitch & complain. It is what it is. Best thing we can do, is try not to give that type things to bitch about. They have nothing else to do, so try not to give them any ammo.
If you are not able to move the coop, or remodel to arrange the pen in such as way that it is not visible, no matter what you do, that jackass will still complain.
Well, I wish you the best of luck. It really sucks when you can't even enjoy what you love on your own property.
(Here are pics with a Pet Gazebo)