Most of us who have gardens* and are fond of nature and animals have outside shelters for them...
'Cathouse' seems off to many Americans because of the usage as a slang term for brothel, and will probably draw a snicker or two..
I participate on Pets.SE, and can find only 1 reference to 'cat-house'
"Cattery" has been suggested on Wikipedia and Quora, but that sounds like a kennel for cats. In other words, similar to an 'aviary'.
'Shelter' sounds like a place for adoption of strays.
Is there a less-suggestive word for the house of a single cat, that works like 'doghouse'?
SWR:My cat lay comfortably dry inside her ______, watching the rain fall on the patio below her.
*My usage of garden here is more the European or Latin-American idea. Garden for Americans often evokes the image of tidy rows of vegetables ripening on the vine and close to hand of the home cook, or a bed of seasonal flowers. By way of contrast, for the rest of the English-speaking world , it usually indicates trees, flowers, bushes, grassy patches, etc.
The perfect environment for birds, rodents, and various creepy-crawlies...
In other words, paradise for a cat.