So some basic math with the following assumptions, the estimate is of sexually mature adults, or of juveniles that will reach adult hood. Balanced M and F population.
1000 lizards. Assume 500 females. 60 babies each. 30000 new lizards a year.
You're going to have to find something to prey on them, or catch a WHOLE lot more than 60.
Presumably something is preying on them, at least the small young, or the whole place would have been overrun already. I think effort in that regard would be more rewarding than driving around checking traps.
Alligators, hawks, storks, herons, egrets, coyotes, raccoons, bobcats, and housecats would all prey on this species as soon as they hatch. Mature ones have fewer enemies but the hatchlings should be quite vulnerable and no doubt a certain percentage do buy the farm.