Boy, oh boy, it's worldwide iconic presence aside, trying to get world heritage status for a structure that is constantly undergoing reconstruction, repairs and uncoordinated pedestrian, foot and car lane planning would be more difficult than getting to the Inaccessible Islands.
I love the bridge, it still astounds me, and I walk or bicycle over it nearly ever day, year round, to get to work, but would this be a tough one. Don't worry, it gets more than enough visitors without the designation, the world knows it's here.