OK, I think I have had a bit of exprience with this sort of problem. My wife is one of those old ladies that thinks she is on a mission from God to save every darn alley cat that comes her way. Of course, when there is a mess to be cleaned up, she doesn't do it. I have to clean the cat pee off of her car windows and just about every thing else.
These are wild feral cats. They are untouchable and vicious. Sometimes I have seen up to 24 of these foul beast show up to be fed.
We put up with crazy people because we love them. But only to a certain extent. No animals are allowed in my house except for birds in their cages.
You need to be very careful when you handle these issues. Sneaky is the word. Fire arms are noisy and dangerous. This job calls for slingshots. Wrist rocket is the best. Keep it out of sight and use it discretely. This is a good use for all of those old marbles your kids left laying around.
When she goes out, it is hunting time at home. Hit a cat with a marble a couple of times, and it will move into another territory. Go mess up someone elses property.
Should one perish, sneak it into the roadway. Then it appears to be a legitimate road kill. Also, disposal becomes a problem for the city sanitation department. Ring them up right away, you don't want it to get high.
As to the issue of the chickens, if she gives in and gets rid of the hens, this neighbor will come back with some other sort of complaint. Your house needs painting, or your car was parked in the wrong place, or any other little petty issued she can find.
The problem is the neighbor. I suggest you minimize your interaction with her. Ask her if she doesn't think the plethora of feral cats is not a health issued? Shouldn't we approach the city about this problem? She won't come back.
Rufus