I don't think it's possible! I've stayed club level on cash and points many times. On rare occasions I've heard they have upgraded or given access but nothing you can purchase.
At AKV (and only AKV), there is a Club Level booking category. There are a very small number of rooms (five 2BR lockoffs) in that category. They are on the 6th floor of Jambo House, adjacent to the concierge lounge. You get lounge access (continental breakfast, snacks and drinks during the day, hot appetizers, desserts and wine and beer in the evening), daily housekeeping and turn-down service.