I think you got it right when you said that it's instinct. Cats are natural hunters and, I'm afraid, there's not much you can do about it. At least, I don't think so. Have you spoken to your vet? Perhaps he/she will have some amazing advice that I've never heard! Bell collars do often work, at least a bit, so I'm sorry yours hasn't done the trick.
Fortunately, my little predator only gets mice and shrews. He's had two birds in his entire life, and never a rabbit or anything bigger than a mouse. I'm lucky, I guess!