In my fortresses I almost inevitably end up building a cat isolator within a first few years. The cat isolator is a single tile wide hallway with a lever at the end, and a door, hatch, and another door leading up to it. The hatch is over a hole in a floor, and has the doors before and after it, and is linked to another lever somewhere else in the fortress. When an immigrant shows up with a cat, I remove all labors on that dwarf. I set the lever so only he can pull it, and then set it to be pulled on repeat. The last door just after the hatch I set to forbid pets. The cat-owning dwarf goes to pull the lever, and eventually the cat decides to bring him a dead vermin, but gets stopped by the pet-forbidden door, standing atop the floor hatch. I then lock both doors and have another dwarf pull the lever controlling the floor hatch.
What is under the floor hatch depends on how lenient I'm feeling. It can be just a single-tile hole in the ground, which the cat will fall into and then never leave. The cat's owner won't get upset because the cat isn't dead, but the cat can't get outside or kill important vermin. Sometimes I'll dump the cat in the caverns to explore them without putting any dwarves at risk. Or I can just put a spike trap or something in the cat dump hole instead.