Same thing happened on the small island of Canna. The rats had overrun the island and were eating the eggs of ground-nesting birds. The only solution was to put down poison in containers that the birds could not get into.
One problem - the Canna mouse, a cute little thing unique to Canna. Where rats can get in, so can mice. Solved by catching as many as possible of the mice and keeping them safe in the zoo until the rats were all gone, then returning them, or rather their descendants. It took a wee while.
Bonus was that the evacuated mice had thrived in the intensive care unit and multiplied, so more were returned than were originally rescued. Also a lot more is now known about the Canna mouse, as they were closely studied.