V.W.
Regardless of the neighbor with the bird... It seems only proper that you figure out a way to keep your cat in YOUR yard only.
Aside from the dangers to the cat itself while out wandering.... they poop in other people's yards just like annoying roaming dogs do. Just because they dig a small hole and lightly cover the top with the pine straw mulch in my flower bed, doesn't mean that I don't end up sticking my hand in it when I am planting flowers. They also jump up on warm car hoods (sometimes they slip a little and leave little scratches in the paint, too)... and leave paw prints all over the windshield. And they cause my dog (who is inside my house) to bark profusely, while they sit demurely knowing (perhaps mistakenly?) that my dog is trapped inside.
I totally understand the difficulty in finding a solution, but it seems the best solution is to keep them inside. Hard to do, once they are accustomed to being able to go out, I know. But it is safer for them and more considerate of your neighbors, if you can't figure a better solution.