The correct answer here is actually E, not D.

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
There are two independent clauses incorrectly joined by a comma.

B. When Charles Lindbergh was attempting his solo transatlantic flight, being very reluctant to have any extra weight on his plane, he
This modifier usage is incorrect; "being" is unnecessary. We also lose the causal structure of the original - he refused because he was reluctant.

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
The "so" makes this a dependent clause, meaning that we have two dependent clauses, but no independent clause.

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
This is unnecessarily passive and awkward; there's no reason for "being reluctant" to be the subject. As a general rule, avoid most answer choices with "being."

E. Very reluctant to have any extra weight on his plane when he attempted his solo transatlantic flight, Charles Lindbergh
CORRECT. We have a modifier, followed by an independent clause with Charles Lindbergh as the subject.

The correct answer is E.