I think the question is "What can't an Alligator Snapping Turtle eat?"
I'm joking, but just barely. In nature they eat mostly fish, but also crayfish, amphibians, smaller turtles, ducklings - just about anything smaller than them. You could possibly have a problem feeding live catfish because of spines, but I doubt it.
I have a Common Snapping Turtle. I feed him mostly pellets and earthworms, plus guppies and feeder goldfish. I have fed him market shrimp and even scallops as well. Catfish fillets are pretty cheap, I may feed him some once in a while.