I am and will be the devils advocate every time on combining prey/predator pets. There are many who have done the combination successfully for many years - that is a fact. There are many who have lost their bird to ferrets, cats, dogs, lizards, snakes or any other predator in the home. You cannot change the nature of the beast, because that is how it survived in it's natural state. Just last week a member had a bird pulled through the cage bars by a dog and killed.
The size of the bird does not make one bit of difference. Cats carry Pastuerella bacteria as a resident bacteria. Birds have no defense against pastuerella and in most cases a scratch, bite, saliva from a cat CAN (NOT WILL) kill a bird - macaw or lovebird. Dogs are dirty in that they harbor many bacteria, fungus and mold spores in their fur and gram negative bacteria in their feces, gut and saliva (that's normal in mammals) and is is abnormal in avian species.
Everyone is responsible for the risk that they are willing to take on behalf of their pets, who are the ones who pay the ultimate price.