As of yet, there have not been any specific predators identified for Brown Tree Snakes.
Few animals have a specific predator.
There are probably several different predators, in the native range, which will eat Brown Tree Snakes when encountered. So yes you could introduce a predator that would eat snakes when it found them, but it would also eat other, non-target organisms. See the following on mongoose. There are two known predators in the world that eat snakes and only snakes. King Cobras in Southeast Asia and a snake from South America, the Mussurana, which is becoming increasingly rare.