The Crown and Fin had quite a few tables big enough for games. All of the booths would work as well as the low tables by the windows. I think you'd also be able to set up a board on top of the chess/backgammon tables if you don't mind sitting on tall chairs.
I'm not sure she the pictures above were taken of the Promenande lounge, but two weeks ago, the tables didn't have the raised section in the middle. (The main tables in Crown and Finn were also tall tables.) In fact, they often used the lounge for crafts and animation so I know the tables were perfectly flat for those activities. They aren't very large though.
In terms of Cabanas, unlike the Dream/Fantasy, that area was completely closed off for cleaning during non-meal times. I didn't try the doors, but there were signs posted that the area was closed.
I think Crown and Finn is your best bet.