Not sure on the Florida Rule for dispatching but in Georgia a bang stick or handgun may be used. That does not mean that it MUST be used. I have used a knife and a a broadhead arrow but this changes the game dramatically at boat side from a safety aspect. Many guides strongly prefer a .22 for dispatch and if you are a do-it-yourselfer, this may be a very good idea. Gators do not like being beside your boat and 200-500 pounds of thrashing lizard at boat side can be a real handful for someone new to that game.
We put two gator arrows in them followed by a harpoon. At that point there is simply no way they are going to get away so our focus is on dispatching as humanely and safely as possible. Shoot me a PM if I can help in any way.
By the way, for you folks that have ever dreamed of this type of hunt, today is the last day to apply in Georgia for a preference point. It is a WILD hunt and worth the wait to get a tag.
Good luck on your hunt!