The Context of Genesis 14 is the early Isin-Larsa Period soon after the fall of Ur III.

I understand why the Amraphel=Hammurabi identification is so popular, but here's the thing, Babylon looms so large in The Biblical imagination that is Babylon specifically had been his royal capital that too would have bene specified here.

Some people in their desire to really sell that identification want to treat Shinar and Babel like synonyms because of how they are sometimes paired together, but they are not.  Shinar is a region which multiple cities as we see in Genesis 10:10.  Now the reading of that verse I favor is "And the beginning of his kingdom was Babel, and Erech, and Accad, all they in the land of Shinar.". But even in the KJV reading this still makes Shinar a region with more then one city in it. 

Ellsar is definitely Larsa, "Er-Akhu" is a known Ephates some kins of Larse had, the earliest know example is only a generation before Hammurabi but it certainly could have gone back further.

The Amorites were already invading Sumer during this period. And that is all the main etymological element Amraphel and Hammurabi have in common implies, their Amorite heritage. 

This period is when Elam was the height of it's power, this alone is when Elam leading a coalition to the Western Levant is plausible. At the start of Hammurabi's reign Elam was powerful within Mesopotamia, but still not plausible to each this far west.