i love the name! i'm sure i would also be confused if i were just meeting her and didn't know your family. i would probably also think caraway was her last name, but if i called her ella and you said, "no, her name is ella caraway," i would be delighted to comply because it's an awesome name (and because i would anyway, even if i thought the name was horrid).
i'm not from the south, but i know or have known many people with double names, can't think of any with five syllable but several with four. i also know many people with very common first names, so they get called by their full names all the time, like celebrities. in fact, i can think of an mdc mama i know irl who gets called by six syllables on a regular basis (and now that i think about it, i want to start throwing her middle name in there and make it eight).
with family, i can see how it would be more difficult to "enforce" them calling her by her name, although i would definitely say, "please call her ella caraway" on a somewhat regular basis. however, i loved all the odd names different people had for me, growing up. it felt like a special secret, sort of like an inside joke, between me and the person who called me by that name.
my maternal grandfather called me missy moo. the missy part stuck for most of my extended family, especially on my mom's side. my two youngest uncles called me louie, and their older brother (the brother nearest my dad's age) called me marilda. my great-uncle called me ethel. my dad called me mischka. my middle school best friend called me mars. one group of my high school friends called me marcie. a random handful of people call me miss m'riss. my name is actually marisa - and that's what most people call me, but i've really enjoyed all the little nicknames. (except missy, actually, i hate being called that as an adult.) nobody is going to change your daughter's name by calling her ella, elle, cara, el-car, ellaway, whatever - it sucks when people don't respect your wishes, but i just wanted to mention that she might really enjoy having a special nickname with certain people in her life (like mil).