I believe introducing some small wild cat population would be a right thing to do, tho I would like to hear opinions of experts.
Edit: I don't think cats are invasive in the way rats are
Not on islands. That's why islands have such unique animals found nowhere else.
And its good to keep rats busy. They won't wander as freely as they do now and hence eat less insects.
The Stephens Island Wren is an excellent example of this http://en.wikipedia.org/wiki/Stephens_Island_Wren
Introducing a foreign species into an ecosystem has failed spectacularly so many times I cannot believe people still propose it.
Snakes or other crawling rat predators would probably be a better idea, but then trying to convince the island people to share their space with snakes and weird looking insects at the same time would probably not go well :)