Yes, That is correct, you replace the fish with the gators. I know for those not from Louisiana raising alligators, seems like a novel idea, but the fact is there have been people raising alligators for their meat and for their skins as a business for a long time. This is only combining the two together. The problem with alligator farms have always been the fluctuation for the price of the skins, this is really where the money is in them. The produce from the aquaponics, would hopefully help stabilize the income in years when the skins were not so high.
Here is the concept. locate your AP business close to a animal processing factory of some kind, (chicken farm, beef, pork processing plant, etc. ) buy the waste meat products, fairly cheap to feed to the gators, and hook the gator tank up to your AP system, just like you would a fish tank.
Now for the hobby AP gardener, just buy some small gators and when they get to big to handle turn them lose and get some more. The game and fish would probably welcome this as in the wild the alligator survival rate for baby alligators is pretty small.