When someone speaks bitchily, they say unkind things about other people — sharp, small, meant to sting, often with a sweet face, or the moment the person leaves the room. It is how you do something, not the name of a person: you can answer bitchily, smile bitchily, talk bitchily. Informal, a bit rude; you would not put it in a school essay or a polite email. Though actually, one sentence can come out bitchily even from someone who is usually kind. It is about that cutting tone, not about shouting, and not about dogs.