I think that the 3rd and 4th nights are the same, but I'm not positive about that.
My advice would be to go on the night of your anniversary. Let them know that it is a special occasion. I personally think people worry too much about which night to go to Palo. If you ask me, Palo is better than any of the rotation restaurants, so it's never a bad move to go there. I always say just pick a night and go. In your case, going on your anniversary night seems to make the most sense.
If you're really worried about missing a restaurant, make reservations for later (assuming you're eating at the main seating) and go to your regular restaurant and have an appetizer or a drink, then eat at Palo later.
As to making reservations before you board, I think you can do this if you're in a category 1-3 suite, but I'm not positive about this. The other way I know of involves knowing someone who is friends with the ship's dining manager, which luckily, I do.....