For the OP, sadly amongst the good advice on here are some rather ill-informed ones!
We've had lots of cats, over many years, a few have collected crashed birds that were learning to fly, but those unfortunate creatures are also high on the menu for magpies and sparrowhawks.
Several liked to catch mice, more to play with them than for eating purposes, we're pretty sure catfood and cat biscuits taste better?
We also have maybe a dozen assorted bird-feeders, some on stands, some hanging from the lower tree branches, the garden birds have a well-developed sentry system in play, checking not just for cats but also the predator birds who come round.
At one point we had a giant, easy-going ginger cat, he'd go out for some air and a snooze, the birds soon got to know him ("he's OK, that's old Ginge") and he sometimes came back with bird poop on him -- he would sometimes have a kip under the tree if he needed shade. But the other cat, went out less often, if she went down the garden the birds immediately put out the alarm call, as they weren't so sure about her.
Our present lodger is a little old lady, and deaf, we think maybe she used to be an indoor cat, she does go out now, if it's not cold or wet, but has only sat and looked at the birds on the feeders.
Check out the local cat rescue places, sometimes they don't know much about their cats, but sometimes they do. Broadly, as someone said, the older cats are less likely to be energetic enough to bother with birds, and male cats are more likely to go and explore a bit. Our large idle ginger boy did - we later discovered - like to visit several other houses in the road to see if any snacks were on offer! But he DID once bring back a rat, as a present. On checking, it had a single wound to the back of the neck, trademark of a sparrowhawk, our hero must have found it and thought he could score some brownie points.
The advice re the very big 'showpiece' cats (Maine Coon, Ragdoll, etc), is quite apt, they're used to being cosseted, not chasing mice and birds.
Oh, and I recall a TV show, 'My Cat From Hell', (USA), in which a behaviourist analysed problem cats. He often suggested changes in their environment, and often reminded the owners that cats need some stimulation, so play with them (NB -- Not ALL cats want to play, some will do it just to oblige their hoteliers).
As I said, go and check the local cat-rescue options (Cats Protection is nationwide, but there will be others), somewhere there's a cat just waiting for your B&B services!