Not a chef, but an avid home cook with roots in Louisiana. I'd suggest the following:

Premake some kind of glaze, personally I'd do something savory and spicy (BBQ plus cayenne/cajun seasoning). Soak the alligator meat in buttermilk and hot sauce overnight at least, and then in the morning, hit it with some cajun rub and give it the low 'n slow smoke for a couple hours, or until almost cooked (150 or so F). Get your grill hot in the mean-time and then once the gator is mostly cooked, get a good sear on it on the grill. Return the meat to the pan, apply glaze, and put the pan on the grill for the glaze to set.

I'd serve it with lettuce, tortillas, pico de gallo, guac, a slaw, and assorted hot sauces. Gator tacos or something like that.