Many critics reproved the curator's bold decision to organize the exhi

• Split #1: The correct idiom is decision TO organize

Besides, "decision that organized" sounds as if the decision is doing the organizing.

Options D and E incorrectly use decision that organized.
Eliminate them.

(Option E also uses "by chronological order," which is suspicious at best.)

Split #2: The missing/wrong preposition

X rather than Y requires that X and Y be parallel.

Prepositions can "distribute" in this construction; whatever precedes X can carry over to Y.

Correct: He voted for Candidate P rather than Candidate Q. ("for" distributes to Candidate Q)
Correct: She reacted with eerie calm rather than visible hurt. ("with" distributes to "visible hurt")

But the preposition that gets distributed must be appropriate for both X and Y.
The word IN is not appropriate for "art movement."

Options A and D (D is already gone) incorrectly use IN:
. . . decision to organize the exhibit IN chronological order rather than art movement.

That usage implies this phrase:
. . . decision to organize the exhibit in chronological order rather than IN art movement.

organize IN art movement is babble. The preposition IN makes no sense with "art movement."
Eliminate A

• Split #3: The better sentence

Options B and C remain.

B. to organize the exhibit in chronological order rather than by art movement

C. to organize the exhibit according to chronological order rather than according to art movement

Two observations. First, although B's prepositions are more common, C's prepositions are not wrong.
-- according to means in a way that agrees with or by. See the second definition, HERE.
-- In particular, in chronological order is more common than according to chronological order.

Second, in the construction X rather than Y, we can repeat the preposition.
Sometimes we repeat the preposition for clarity.
In option C, though, repetition of the same preposition creates a longer, bulkier sentence than is necessary (witness: option B).

We cannot use IN for art movement. Although a single preposition will distribute, in B the sentence rightly uses different prepositions to
suit each item.

I would decide the matter on rhetorical effectiveness and style.

Option B is more concise. It uses two short words rather than two long words plus two short words.
Option B also shows more contrast by using two different prepositions.
Although that usage is an accident because "in" does not suit "art movement", nonetheless the different prepositions create a contrast that is consistent with the meaning of the sentence and with X rather than Y.

That is, B wins the style and rhetorical effectiveness contests.

Answer B