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 if Babylon specifically had been his royal capital that too would have been 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 with 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 epithet some kings of Larse had, the earliest known 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 at 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 still powerful within Mesopotamia, but not enough to reach this far west. 

There is also the record of an Abuha Son of Ishmael during the reign of Hammurabi. This Abuha must be actually a grandson at least since that's not the name of any of Ishmael's sons in Genesis 14. So generations separated Abraham from Hammurabi.