The cat mint idea is worth trying. It is easy to grow and unlike other mints it doesn't take over your whole garden. If you plant it away from the run there is at least some chance the cat chooses to sit near it instead of going into the run.Nice idea but: No, I don’t want a big rooster with my bantams. They are just too heavily shaped for the tiny bantam hen bodies.
And in my opinion its never worth the risk to add a temporary newcomer with possible diseases in my flock. I think its forbidden too nowadays, because we have serious HPAI restrictions.
A cat in the run is no big problem, just a nuisance for the chickens and little songbirds that visit the run too.
Not all cats respond to cat mint but I believe most do. I once had a cat that would lie under the cat mint in some kind of drug coma (though I admit it is sometimes had to tell with cats). He would lie on his back with his tongue lolling out!