The original poster was uncomfortable killing the cat. He probably has the right if covered in the laws of his area. That's not the point.
He has to protect his animals. Cats know no borders (my yard, your yard). His neighbor is not getting it and does not want to have to consider a lifestyle change for her cat. Without her cooperation it will lead to having no cat.
This solution will work for now, but sounds like it will lead to problems down the road in terms of harrassment, suits, more cats, etc.
In my book killing is the last resort. It may come to that, but there are options before getting there. It just takes a little time to work out solutions like this. It's not a wild animal, it's a pet, with an owner that needs to understand how to protect her cat. If coyote's were roaming your neighborhood, would she let the cat roam at large?
The paint ball was one kind of notification. The neighbor took that as assault on cat, not as 'warning, cat in dangerous to cat territory.'
Right now this is only a problem for the poster. The cat lady doesn't seem to get it is a problem for her as well. You have to make it a problem for her for her to start paying attention.
Trapping the cat and taking it in to animal control, or calling animal control to take it in are also an option. I like this one because it starts to make this a problem for the neighbor, not just a problem for the poster. She will have to pay to get her cat back each time this happens. It may take a few times for her to wise up.
She needs to research ways to keep her cats away. That may mean it becomes an inside cat, or stays in her fenced yard with an electronic collar or declawing so it can't climb the fence, or she may need to rehome it since she is letting it be in danger like this.
In the meantime, good protection on the poster's part, good fences, covered runs, good enclosures and keeping the birds 'in' til it is resolved, will help keep bird death down.
And a big mean rooster to patrol the grounds.
That's more than 2 cents, it's my 2 cents and change.