Our life list of birds inside the house is 4, a chickadee, also a titmouse, a nuthatch, and a downy woodpecker. All brought in by the cat for something to play with, so still very much alive (though perhaps a bit unnerved) when they got out.
It would be nice if cats took the mice out - at least in return. But they seem to want to proudly show off field mice caught, too. And maybe small snakes even. It has been years since Scotty died around age 18, buried along with Pooka on a knoll out back.
I could enjoy a rent-a-kitten, who visits for short whiles, for smiles.
We would have been happy to rent Bob, our cat. He brings home mice, voles, moles, chipmunks, squirrels and birds. But none of these get out afterwards - mostly they are either 50 - 90% eaten (some organs left on the carpet), or else pretty much dead after a good sporty play. Usually happens at night without spectators present.
All those birds, but no bats? We get one or two every year that eventually get chased back outside.
Ernst - in some cultures they eat cats! maybe yours is such a candidate:-). Unless, of course, you find it irresistible & lovable, if that's possible with cats. There were times when I quite enjoyed Scotty & Pooka.