Needless to say, the title refers to the Hebrew bible, about which I will now proceed to speak in highly hypothetical, but hopefully generative, terms. That central to the Bible is a covenant between God and the Jewish (let’s set aside disputes over this name for now) people is uncontroversial; that the entire Bible is nothing more than an ongoing consideration of the history, consequences, implications, terms, identity of each partner, etc., of this covenant might be a bit more so, but probably still mostly plausible. But if we get more precise and, we could say, more “realistic,” about the origins of this covenant (following the suggestion of Bernard Lamborelle) and say that the covenant is in fact a title deed to a piece of land granted to someone who came to be called “Abraham” as part of an allocation of responsibility for supervision of the parts of an empire by the emperor (or his delegate), we enter into more unknown territory. For one thing, this approach makes the entire affair far more political, because if God gave your people the land in perpetuity that would be an end to it,you’re your rights would be secure; if, however, the memory of a more mundane exchange is never completely lost, the claim to the land must constantly be re-asserted. (As an aside here—but an extremely important point to return to—this hypothesis also helps to clarify another ancient Judaic innovation: entering into a covenant
I would correct your paraphrase of my hypothesis by saying that Abraham "exchanged" his sister for Pharaoh's or someone in the royal court, receiving Hagar (being Egyptian) or Keturah in return (who is often theorized as being the same person as Hagar).
In brief, the Abraham story seems to be a merge of 1) a patrilineal descent (PLD) story where Sarai is marriageable to Abram and 2) a matilineal descent (MLD) story where she's Abram's classificatory sister and therefore unmarriageable. I believe the ambivalence over whether she's his sister or wife stems from these two accounts being merged together, likely during Babylonian exile as a deal between the priests (Levites) and the princes (Judeans).
In the PLD version of story, Abram and Sarai are marriageable, not siblings. Note that under ancient kinship, a "sibling" is simply someone of the same generation who shares your clan designation, and under exogamous rules such siblings are unmarriageable. However under PLD/endogamous rules, they can share a clan designation and their marriage be licit (possibly even required). Such marriages might be between first or second cousins (and the Catholic church extended it to 6 cousins), but whatever separation is required to be marriageable under PLD, we can assume they met that requirement. To an outsider under MLD, they would have been considered "siblings" by their shared clan designation, and they might have even appeared to share a "father" (which would simply be the same clan designation of the older generation), but they would not be genetically siblings. Under this story, Abraham's inheritance passes through the male descendants to the 12 tribes and finally to Judah. This narrative benefits Judah who claimed a right to the Davidic throne, so I would assume it's from the E or J doc. It also benefits Christians whose Messiah is the patrilineal (and spiritual) heir of David.
The 2) MLD version of the story only works if they share a mother. I'll explain later. In this altered MLD version of the story, Abram and Sarai are siblings who share a mother, and Abram gives her to Pharaoh to form a political alliance, which means Abraham's inheritance transfers to Sarah's offspring, who are presumably born into the royal court, and likely carried Egyptian names: Moses, Aaron, and Miriam, who leave Egypt to stake their claim (and being priestly, Pharaoh obviously tries to prevent this exodus). This narrative benefits the Levites who then use it to justify their claim to Israel, so I would assume it's from P. This version also makes sense of Hagar's son Ishmael's inheritance of all of Arabia, since such inheritance would have passed from Pharaoh, and it also makes sense of YHWH's prognostication of Israel's 400 year sojourn in Egypt, which only works in this timeline, but not in Jacob's (which is only half as long).
The reason that the MLD version falls apart if they share a father is that, in that case, they would still likely be marriageable, unless it was a 2-class system in which case they would have adopted the same clan affiliations as their mothers (who would've likely been siblings). We need to posit the most likely scenario where they are classificatory siblings to justify the ambivalence of the text, hence my belief that they shared a mother in the Priestly, MLD version.
The only way to really merge these stories was to 1) settle on Abram and Sarai sharing a father, 2) make a mess of Abraham's story in Egypt, and 3) somehow get Jacob back into Egypt.
The PLD version suffers from two major flaws. 1: Isaac. Yoreh's theory that Isaac was actually sacrificed might be true. How else do we explain the shoddy writing around Isaac's life? There's a copy-paste job where Isaac apes Abraham's Abimelech story as well as the story of the wells, and the rest of it is pretty uncompelling and seems to just be a political dig at the Edomites who had been moving into Judah during Exile. The other flaw is 2: Joseph. Beautiful a story as it is, as you said, it plays as more a polemic about tribal bickering over the bordering land of Benjamin.
The MLD/Priestly account is far more compelling. I would argue either that it was the more complete of the two, or the redactor chose to keep its contents as close to the original as possible in order to justify the succession of the priesthood. The one shift it would have to make would be to change it to PLD to justify Aaronic descent, and this might have been a deal they had to make.