I’ve had that problem in the past in my greenhouse. I locked my ‘Killer Queen’ cat, Eva Grace, in the greenhouse overnight and...problem solved. Word must’ve gotten out to The Mouse Community, as it was no longer a problem after that. ;)
We’re allergic to cats and this is tick country.
I cannot imagine the amount of effort it would take to keep an outdoor cat anywhere near tick free.
So we use conventional mousetraps and Top Gun bait blocks.
It has something in them that messes up their digestive tract and they stop feeding but it is pretty much harmless to anything bigger or that might catch them for food.
https://www.jteaton.com/product/top-gun-bait-blocks-4-lb/