I've never killed a gator, but I do know that a big gator's brain is about the size of a walnut, and it is encased in a very thick, heavy skull. Guys kill gators with shotguns all the time, but it's not an easy trick, apparently. I guess you immobilize them with rope, shoot 'em, and let them bleed out.
I've killed big snapping turtles, and those things are so mean even chopping off their head doesn't kill them right away. They'll thrash around for a good while, and the head will continue to bite even after you cut it off. Cut them open and remove the heart, and it will continue to beat for several minutes. Big reptiles die hard- they've been around since the time of the dinosaurs and their design is so good that it has remained unchanged since then.