What a lovely, lovely dress. You know what? If I were you, I would pull out virtually every necklace, scarf, jacket, and cardigan I owned and spend a good whack of time in front of the mirror, trying every wild, weird, and wonderful combination I could think of. I mean, the whole point of a LBD is the versatility, right? Although it's true that yours is so beautiful on its own, it does skew more to the formal side.
But I don't see any reason at all why you shouldn't vary the accessories according to mood and event, rather than trying to find a "perfect", one-size-fits-all combination. Pick your favourites and don't rate them as better or best, but as "elegant", "playful", "more casual", or whatever.
Have fun! I can hardly wait for the day when I find my LBD!