The Duchess invites Oliver to her estate because she knows that he craves social respectability. Oliver may well be a fantastically wealthy man, but due to his humble background he's still looked down upon by the upper classes. He has everything he could possibly want in life except the social acceptance by the elite which he believes is his due. The Duchess knows this, which is why she plays upon Oliver's obsession with social climbing and invites him to her estate. She's hoping that he'll be so enchanted by the trappings of high society that he'll be more likely to help her out of her sticky financial situation. It's a sign of just how desperate she is that she's prepared to invite someone over who ordinarily wouldn't be allowed through the front door. But desperate times call for desperate measures.

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.