Even though her career was cut short when she was in her prime and the fifteen recordings she made were disappointing artistically as well as technically, Olive Fremstad (1871-1951) has never been entirely forgotten by opera aficionados.

her career was cut short in past..wherever possible simple past is preferred. And "when" is a better choice when we want to use simple past ..

(A) though her career was cut short when she was in her prime and the fifteen recordings she made were
Cream of the crap...Best answer choice....

(B) though her career was cut short while in her prime, with the fifteen recordings she made --> out because of reason explained above

(C) as her career had been cut short when she was in her prime, with the fifteen recordings she had made
There is no reason to use "had" to show a sequence of events..Its already implied in the sentence that her career was cut short when she was in her prime...Sequence of events is already implied...---:(

(D) with her career having been cut short when she was in her prime, and the fifteen recordings she made were
-- wrong sentence correction...even with her..

(E) with her career cut short while in her prime, and that the fifteen recordings she made were --> out because of reason explained above

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As conjunctions (words that join parts of a sentence), when and while both mean 'during the time that' or 'at the same time as' and can sometimes be used interchangeably:

(1) When/while I was cooking, I heard a knock on the door.
(2) She lived in the dorms when/while she was a student.

While is always a better choice to talk about actions that are (or were) taking place at a specific point of time in the present or past (therefore used in the Present Progressive/Continuous and Past Progressive/Continuous tenses:

(3) He tripped over someone's foot while he was carrying the tray.
(4) While the sun is shining here, it's raining up in the north.
(5) While I was waiting for the bus, two teenagers were smoking right in front of me.

When is possible in examples (3), (4), (5), but less preferable than while.

When is the only choice to use with the Past Simple tense:

(6) When the bus arrived, I showed my ticket to the driver.

While is not possible in sentence (6).