Looking for an old school Rat trap.

The lesser of two evils is still not a great choice, but yeah, I'm with you. Why not simply have a cat?

If a rat had a choice, I suspect it would prefer having its neck broken in a trap, rather than being eaten alive by a cat.

Where there's one rat there are usually a few dozen more. They will avoid the cat altogether and go back to the woods where hopefully a nice snake or raptor bird will find them.