I didn't realize that Moochie was underage. But I still say "SSS"....but use a paintball gun. Might get the owner of the cat, assuming there is one, to start asking questions around the neighborhood. THEN Moochie can tell the owner what their loose cat is doing to the birds.
With that said, I will tell you from experience, that if a cat is used to being outdoors, it will NOT stay indoors. That is how I got my cat. It tried to escape every time my mother opened the front door and she wasn't able to catch it every time. So it came to my house where it can come and go, at will. It is much happier here.
Which leaves the only other alternative. Predator proof the chicken's enclosure. Which is far more legal than shooting a "pet" and pays bigger dividends in the end.