One day in NO- breakfast, lunch and 2 dinners?
For your anniversary dinner, I would suggest CP as the most romantic space (and great food as well). Be sure to let the restaurant know the occasion when placing the reservation, and they will add a special touch or two.
For the other dinner: Arnaud's or Galatoire's
Lunch: Mr. B's (Gumbo Ya Ya and/or BBQ shrimp)
Breakfast: Stanley is a good pick. Others include Atchafalaya in the Irish Channel and Elizabeth's in Bywater.
I'd have breakfast at Stanley!, take a long, winding walk, head to Mr. B's for gumbo, take a nap, proceed to Arnaud's bar for a French 75, then feast on Creole favorites. Since you're in the FQ, my 2nd choice of these would be Galatoire's. A 3rd to consider, more for romance than food, is Broussard's. Also, for a drink before or after, maybe hit Patrick's Bar Vin, also in the FQ. Have fun!