I would call no ifs ands or butts about it.

Sure it would be terrible for a cat to be euthanized if it was healthy and friendly but just what if it isn't? You've got a childs life in your hands.

BUT I would also offer to quarantine the cat for 10 days (you can do that instead of killing the cat to test for rabies).

But if I had a stray cat in my hands I wouldn't let it go again no matter what. I'd at least take foster it or take it to a shelter or vets.