I have a shed cat, as the office is above the garage so to get to the house area they have to go through the shed.
You might need 2 cats if the workshop is on the same level as the house
Get a pair of unwanted kittens from one of those do-godder groups de-sexed females if possible
put them in a big dog cage ( you can rent them ) , one in the shed & one in the house .
Put some old unwashed cloths in the bottom of the cage and regularly pet the cats so that they bond with you .
After a week or so open the door and let them out but keep on feeding them in the cage
Then dump the cages all together .
Reduce to "play time " so they get bored .
Start to feed them only in the morning so at night they are bored & hungry.
Cats will eat spiders, worms, mice, lizards beetles , cockroaches and most anything that moves.
So the trick is to lock them inside from dusk to dawn so they will clean up the shed & house then give them a big feed of dry cat food in the morning when you let them out ,
That way they will be full during the period that most of the wanted wild life is active and do the least amount of damage.
This is how your ancestors managed cats befoer the pet industry developed to use them to fleece the owners & bleed them dry .