Why does the Duchess invite Oliver to her estate?
In my opinion, the Duchess does this to try to distract Oliver Bacon and to make him more likely to accept her "jewels" and pay a high price for them.
The Duchess is really in a desperate situation and needs the money badly. She knows her pearls are fake but she needs a high price for them. She knows that Oliver wants to be included in high society -- to be seen as an equal to people like her.
So as he inspects the pearls, she invites him to spend a long weekend at her place. She hopes (and gets her wish) that this will really appeal to him and he will not inspect the pearls too closely.