Stray cat is a vicious hunter

Our neighborhood has a lot of strays. We leave food out and regularly feed three of them. There is one in particular we have been feeding and gaining the trust of for over a year. In the last few months, he’s grown to trust us. He’s super affectionate and cuddly and we were able to get him into a carrier easily and to the vet to get checked for any medical problems that could pass to our indoor cats and got him his vaccines. Once we had the all clear, we started allowing him into the house because it is winter and very cold. He sleeps inside at night, but as soon as my dad leaves in the morning, the stray (Shadow) will run out the door with him.

I am fine with him being an indoor/outdoor cat since the outside is all he has known (I know this is an unpopular opinion). However, this cat is a hunter. On multiple occasions he has brought mice and a bunny once into our garage and EATS THEM. I am very worried he will catch something from eating these animals and pass it to our cats since we let him inside.

Honestly unsure what to do. I think it would be impossible to keep him inside 24/7 because my dad is in and out all the time and Shadow just slips right by him. But, I really don’t want him eating any animals and catching something. I would hate to stop letting him inside suddenly, but want to protect my indoor cats. Has anyone experienced this before or has advice?

Please be kind, I don’t have experience with strays/outdoor cats besides feeding them.

NOTE: he is fixed, on dewormer, and gets along with my indoor cats and the strays in the neighborhood