Humane? Some wild animals eat their prey in chunks while it's still alive, how is that humane? Nature doesn't worry about what's humane. It's entirely up to us whether or not WE do.
I think there's a line between feeling bad about having to kill something to feed a pet (which seems a pointless thing to feel bad about, to me), and absolving oneself of all responsibility for a death because the pet did the killing. But, your mileage may vary.
If the screaming bothers you, don't feed live.