Okay, here use of idiom "Dispute over" is the cjeck point. Lets short list all valid choices...
A and C.

B doesnot make any sense to me. E is worng idiom. Same stands for D.

C...the use of "There was" makes it wordy...

Thus, A is correct.

Option A.
This option makes a proper use of the idiom 'Dispute Over' and clearly states the reason for the break up without any too wordy explanation .This option is the correct option. It follows a proper parallelism in a perfect order.
option B.
This option sounds as useless and it reads. 'Over bitterly disputing' It sounds awkward and it does not look suitable.Thus, this option is incorrect.

Option C.
Wait! This option looks correct but let's look at what separates it from the correct option above which is option A. the use of 'after there' in option c and the use on 'in' in option A is the significant difference between the two. So for safety and proper grammar, Option A would be considered over Option C.

Option D.
The usage of Dispute should be clearly seen that it is 'dispute over' and not 'dispute about' that is the correct use of the idiom. This makes option D invalid.

Option E.
Where's the 'over'? It should join the 'dispute' to express the 'break up' as wanted by the examiner. Like it was established previously, 'dispute over' is considered over 'dispute about'.

Hello!

Instead of going through each option, I'll stick to answering your specific question about why answer C is wrong, and A is correct.

Answer C states "after there was a bitter dispute over," which, if you read it carefully, isn't clear as to WHO was arguing. Did Camus break with Sartre because the two of them were arguing? Or was it because two other people were arguing, and the two of them were grouchy about it? Just saying "there was a bitter dispute" isn't clear enough for readers to know the full story.

Answer A works best because it clearly shows that Camus split with Sartre AS the argument was going on, and that they were the two arguing. There is no way to misinterpret this one, so it is the best option.

I hope that helps! I'm available if you'd like any follow up.