Just in case you haven't thought of it, there are other names that easily take Eddie as a diminutive:
Edwin
Edmund/Edmond
Edgar
Ed, uhhh, I ran out of readily accessible names, unless you want to name yourself after a cheese.
And variations on Edward--Edouard, Eduardo, Edgardo, Ed Cetera.
I was thinking that Edward Elliott James G... sounds damn good (depending on your last name), but that would mess with your initials. Or you could still use the same initials if you really want to, couldn't you? Hey, why not?
You do look like an Eddie, so that's a good choice. But I favor taking both Edward and Elliott. (If they were different names, you might be able to get away with hyphenating them, but I don't think that will fly here. Then again, what matters is what YOU like.)
P.S. I also thought you could do some kind of hybrid name, but it would sound too exotic (and thus call attention to you). Besides, my first thought was Eddiott, which is just plain silly. And wouldn't sound so hot if people didn't pronounce it just right.