[Quiet Man Cometh]

Would I be right in assuming that both dogs are female? Females will often scrap with each other to establish pecking order. It could be that Lizzy feels that when Duchess is outside, she has the change to "take over" the house and is playing dominant by not letting her back in. If she doesn't harass the other dogs, it could be that she knows they are already submissive. Since you mention that Duchess isn't the type to put up with nonsense, they might have unresolved dominance issues.