Yes. Even then. Such a dog should be put down.
Put down? I”d give the pooch a medal. You haven” t seen what can happen when several hundred feral cats infiltrate a neighborhood. They are nasty smelly destructive pests, and not the cute kitty that deserves love and attention. Dom”t be so naive.
No the dog should not be put down for doing what dogs are bred to do.
I hope your cats are all house cats because as my vet says “outside cats live short violent lives”!