The trick here will be to find something the snakes eat that alligators do not. As has been done on Guam against brown snakes, acetaminophen (generic Tylenol), is deadly to reptiles (and many other animals), so they stuff dead mice with them and it kills the snakes.
However, lots of animals will eat dead mice.
Maybe it kills them because it lowers their body temp a little. Are alligators quite as fussy concerning cooling as these snakes are, I wonder, as alligators have survived many a colder snap in Florida throughout the ages? Other reptiles live in colder places and survive. I am not at all sure that Tylenol kills them because it lowers their temp, just a thought.