Charles Lindbergh, for his attempt at a solo transatlantic flight, was very reluctant to have any extra weight on his plane, he therefore refused to carry even a pound of mail, despite being offered $1,000 to do so.
(A) Charles Lindbergh, for his attempt at a solo transatlantic flight, was very reluctant to have any extra weight on his plane, he therefore -> IC, IC is incorrect structure. Incorrect.
(B) When Charles Lindbergh was attempting his solo transatlantic flight, being very reluctant to have any extra weight on his plane, he -> "being very reluctant" is incorrect.
(C) Since he was very reluctant to carry any extra weight on his plane when he was attempting his solo transatlantic flight, so Charles Lindbergh -> Since...so is incorrect construction.
(D) Being very reluctant to carry any extra weight on his plane when he attempted his solo transatlantic flight was the reason that Charles Lindbergh -> His solo transatlantic flight was the reason or his very reluctance to carry any extra weight. Incorrect.
(E) Very reluctant to have any extra weight on his plane when he attempted his solo transatlantic flight, Charles Lindbergh -> Modifiers in place and we have other errors fixed. Let's keep it.
So, I think E.