Sure, I don’t think anybody disputes that the Bible is a not single book written at once.
But the claim that any part of Genesis dates to the Exile is a weak historiographical claim, based primarily on a few pillars of very weak circumstantial evidence: a) we don’t have many physical examples dated to a prior time - but that’s not at all surprising; the vast supermajority of texts don’t survive, and many of the most ancient copies we do have are relatively recent discoveries; b) it makes sense to some people to argue this based on (reasonable) speculation about the development of the Jewish religion and the pressures at different periods of its development and c) as a corollary to b, one mainstream opinion is that Genesis can mostly be attributed to two different sources by a stylistic analysis and one of them is simply believed to date around the Exile.
This corresponds to the general current fads in historiography and history but it is by no means definite, and we shouldn’t be surprised if it is all overturned in a night by a single discovery, as these things very often are. This type of historiography is not really rigorous in a meaningful sense and is based essentially on current academic trends and a few authorities.
So we can safely consider the arguments underlying the dating ourselves without getting in too much trouble, but we don’t even have to do that to address your particular claim that the portions of Genesis must have been devised by an Exilic writer due to Babylonian influence. This essentially requires engaging in a deliberate pious fraud, which then was taken up by the other Israelites without question and who accepted the new stories with no recorded debate, and which also received no pushback when the Israelites who did not go into Exile. We would expect the possibility of diverging traditions on this point, but for example, the Samaritans have precisely the same story on this point, although the dating of that is also a point of contention. Either way, we should keep in mind that only some Israelites went into Exile, and plenty remained in Israel and Judah and maintained their traditions through the 70 year Exile.
It’s possible that this did happen. But it’s also the case that Babylonian contact was not new in the Exile period. The Israelites didn’t exist in a vacuum, they were part of the tapestry of the wider Semitic world for their entire existence and would have had familiarity with all these stories and beliefs the entire time. Indeed, it’s most commonly believed that the Israelite religion was not unique, but a simple development into monotheism from a normal localized Canaanite religion. And indeed, at some point, they all shared a common origin. Rather than adapting a Sumerian-via-Babylon story in a very late period, it’s a much more parsimonious explanation that some version of the story always existed amongst the Israelites in their oral tradition regardless of when it was committed to writing.