Mia. So your neighbor stated chickens OK at first then when It was time to get a letter signed they go south on you correct?
If so you might fight this.
Media works wonders for common sense to prevail.
Show them the expense etc you went to after receiving neighbors permission.
I know the argument back at you will be but it was your responsibility to get a letter and my reply back would be I took their word on the situation thinking they would keep it like a DECENT person would.
It's worth a shot but legally that neighbor and city council who drew up the joke have you any way you look at it.
It is beyond my belief why anyone should have a right to say what you have, or what you do on your property, until such action becomes a harm to another persons quality of life and when that happens meditation is the way to solve it, not blanket ordinances by power tripping so called educated fools who have never read the Constitution!
You might try going down to city hall and see what the procedure is for getting an extension.
If you say you are looking for homes for them hopefully someone down there not power tripping will give it to you.
Then I'd spend that extension time moving my run.
Giving up?
No go. NEVER!
Why not move the run and install a different door entrance to it?
Home Depot has the gate materials.
If money is tight forget the more expensive gate and cut the door out.
Cement in two cherry timbers each side of door.
Make your door using the thicker then chicken wire square wire.
Take 1x2 wood around perimeter of wire to sturdy it up and you got a door.
Many ways to attach it.
Simple hinges will work.
Make a couple of cross bars attached to cherry timber to go across for security.
Do what you have to do to keep them.
You can do this.
Don't let some one whose life is so miserable the only joy they get is to power trip on their neighbors.