I am guessing a lot of us would have incorrectly marked E because we are drilled with the dictum "if you see a gerund, eliminate it" during our GMAT prep. But let me make sense of this here.
(A) of being singers who must also perform as dancers -- "being singers...perform as dancers" is a noun entity here. Sounds correct and makes sense.
(B) of singers who must also perform like dancer -- Incorrect. difficult task of <<what??>> singers? It is missing a verb here. Perform like singers? Be like singer? Its unclear.
(C) that they are singers who must perform like dancer -- Incorrect. The "that" here is problematic and unidiomatic.
(D) that, as a singer, they must also perform as a dancer - Incorrect. same as D.
(E) to be a singer that must also perform as a dancer -- Close. But this is worded incorrectly. "to be X that must also do Y" could have been better conveyed as "to be X who also does Y". In light of A, I would eliminate E.