As I understand it, there are two ways to do it here legally. First is to be a contract hunter called on to do the deed once the gator is declared a nuisance. Those guys usually work in daylight, bind the gator,pull it ashore and do the job at point blank with whatever is at hand. Generally, this takes place with people around, so the slug, bullet, or whatever must stop in the gator.
Second way is to win the lottery for a gator tag. These hunts take place at night, usually usually in isolated areas, by boat, and the tag holder must be accompanied by a licensed guide. My choice in that hunt would be something like a 376 Steyr, short barrel, compo stock, 2x scope with illuminated reticle. That, or whatever the guide says.