If you have the kind of
greenhouse that has a door, I assume you are keeping the door open or keeping the cats inside.
Keeping the cats outside will do what you want the cats to do.
We have a trailer that the mice thought was a good place.
In May of last year, a cat left her kitten with us and did not come back for her.
We no longer have a mouse problem. The cat never goes into the trailer.
The thing about outside cats is that they tend to wander ...
They do come home to eat if fed frequently.