Not aware of any Invitation Only restaurants except perhaps the Captains Table.
All I ask is a tall ship and a star to steer her by; John Masefield
Carnival: Glory 2004, Destiny 2008, Splendor 2009, Freedom 2011, Valor 2012, Dream 2013
Celebrity: Summit 2011
Princess: Ruby 2010, 2014, Caribbean 2013, Coral 2014, Regal 2014
Star Clippers: Royal Clipper 2015
Chef's Table, which is fabulous. It is held in the library, though it may be more $$$ than you want to pay. 7 course dinner and a kitchen tour and includes unlimited wine. So, if you are not a wine drinker, not worth the $$$.
For a while Carnival was testing a Fee-For-Steak on ships that didn't have a steak house, like the Carnival Ecstacy. Prime cut steak for a fee of $18 could be purchased in the main dining room. It is cheaper than the $30 steakhouse charge, but of course you don't have the atmosphere that you would in the steakhouse. I'm not sure if this is still an option but it did give you an opportunity to order a nice filet mignon instead of the flat iron steak.
Enjoy your cruise,
Ron and Sue
Carnival Sunshine - January 2014
Carnival Dream - January 2015
The Prime Steak option is still available on some ships, but I don't think Ecstasy was one of them. The $75 Chef's Table option is available on all Carnival ships. Kind of steep, but it includes a number of courses and also wine.