Or if they inherited it from earlier sources, such as byzantine law or the pre-schism church, when did that ban start?
I know that the Justinian code didn't ban cousin marriage, and that the later Ecloga (Byzantine law ~726) and Kormchaia (Church law ~1050s) did, but my questions is regarding whether Church immediately took the position of banning it as was under Byzantine law (or even Byzantine law following church doctrine) or whether it took them until later.
Not an answer, but perhaps a crib for further research
banned by the Christian emperor Theodosius I in 381 in the West, and until after the death of Justinian (565) in the East,Wikipedia:cousin marriage citing Ottenheimer and Grubbs \
However,
Since the 13th century a ban on marriages between close relatives was formalized by the Russian Orthodox Church in nomocanon referred as Kormchaia. Kormchaia stipulated written rules on how to determine which marriages were invalid. Generally all marriages that resulted in crossing bloodlines were prohibited. Wikipedia:Prohibited degree of kinship citing Saharov
I infer that sometime between 381 and the publication of the Ecloga in 726
I would speculate that the restriction originated in the Ecloga because of this quote:
The structure of the act is original and it isn't taken from any other source, considering that Leon didn't want to complete layer legal reform. It seems that his goal was just to modify Justinian's legal tradition in the most important segments of legal life, while still adapting it to the needs and actions of the Middle Ages.[citation needed] It needed to be distinguished from its original model. . . . Patria potestas is decreasing its power influenced by Hellenistic and canon laws and the rights of women and children are increasing. Tellegen-Couperus. Ibid
I am aware that much of this recapitulates research that is implied in your question, but my hope is that attaching dates and citations will make it easier for someone else to find a more precise/specific answer.
It is also possible that the answer is simply, "We don't know (yet)"
