Introduce some feral cats.
they've been there 200 years. they're as natural as anything else now.
That's not true. Those birds have been using that ecological niche for millenia, the rat's have been there but 200 years. The rats evolved in a different area altogether. Natural populations in an eco-system do not disrupt the balance like these rats will.
Nature hasn't made a mistake. Animals jumped ship from somewhere else and settled on these islands. They aren't meant to be there, only through human transportation as a vector are they found there. Like other invasive species, zebra mussels, green crab, tunicates, spruce beetles, purple loosestrife, etc.