On this page we attempt to make available the experience of individual bicycle tourists who travel to Libya (you can share your experiences here).
Libyan Arab Airlines
Libyan Arab does not charge for bikes in a box Benghazi to Tripoli and Tripoli to Dubai: return. Nor do they seem to have weight restrictions....for that matter they don't care if you smoke on their "non-smoking" planes.
No damage and no hassles.