The correct idiomatic expression for this example is choice E - as a child. Choice D's at the time of indicates a brief amount of time, certainly more brief than childhood, and is unidiomatic.
While being a child (B) seems to imply that John Doe has been acting like a child, rather than referring to his actual childhood.
Choices A and C are simply not idiomatically correct.
The correct answer, choice E
, fits with as from our list of idiomatic expressions - as a child follows the same format of as a scout.
"Start By Doing What Is Necessary ,Then What Is Possible & Suddenly You Will Realise That You Are Doing The Impossible"