If you happen to have bought practically every Disney movie ever made in the last couple of years (and still have all your receipts), you can buy a private meet and greet with a character with Disney Movie Rewards points.... but you have to have LOTS of points (15,000 to be exact). Since you only get 75-100 points per movie, you can only use each title once, and points can't be combined from multiple accounts, it is unlikely that is a reasonable option, but you never know....
I would think that Ariel's Grotto would be your best bet. There is a little cove Ariel meets her guests in so it will be semi-private (there is usually a Photopass photographer there as well as a character handler). It is more private than most meet and greets as you won't be visible to others waiting in line (the next family behind you will probably be able to see, but that will be all).