When did Hammurabi live?

One of my most viewed posts is Dating of the Hammurabi Code. In it I propose a revised dating of the code based on the premise that many dates are tied to Egyptian dating which I think is incorrect. My reasoning was as follows

• The very earliest date for the beginning of the Egyptian dynasties is c. 2200 BC
• Moses led the exodus of the Israelites out from Egypt around the time of the 12th and 13th dynasties (which may also correspond to the 6th dynasty)
• The 18th Egyptian dynasty was of some duration. The beginnings of which are possibly about the time of Samuel and Saul
• 1750 BC in traditional Egyptian chronology corresponds to the 12th and 13th Egyptians dynasties
• The first Babylonian dynasty ended with the fall of Babylon is dated c. 1500–1600 BC and corresponds to the beginning of the 18th Egyptian dynasty 

Moses led the Israelites out of Egypt c. 1450 BC. Saul began his reign c. 1050 BC. Thus I believe end of the 12th and 13th Egyptian dynasty to be out by ~300 years and the beginning of the the 18th dynasty out by ~500 years.

Hammurabi is dated variously to 1700, 1800 and 2000 BC. Shorting this by 300 to 500 years gives a range of 1200 to 1700 BC. Thus he could be contemporary to or even post-date Moses.

It is more complicated in that the 6th Egyptian dynasty may be contemporary with the 12th and 13th, and the assigned Babylonian dates may not completely rely on Egyptian synchronisms (as per my assumption).

Since writing this I have come across a discussion of Abraham's war with the northern kings. In Genesis 14 we read that

At the time when Amraphel was king of Shinar, Arioch king of Ellasar, Kedorlaomer king of Elam and Tidal king of Goyim, these kings went to war against Bera king of Sodom, Birsha king of Gomorrah, Shinab king of Admah, Shemeber king of Zeboyim, and the king of Bela (that is, Zoar). All these latter kings joined forces in the Valley of Siddim (that is, the Dead Sea Valley). For twelve years they had been subject to Kedorlaomer, but in the thirteenth year they rebelled.

Shinar is the plain where the tower of Babel was built and the location of Babylon. Amraphel was the king of Babylon. Amraphel has been identified with Hammurabi for over a century. Phonetically we have

• (a)-m-r-(a)-ph[p, b]-(e)-l
• (ha)-m(m)-(u)-r-(a)-b[p]-i

That is mrp-l and mrp. The same name other than the final el, Hebrew for God. A designation he likely used as Hammurabi was deified.

Abraham's battle with the kings occurred between his departure from Haran and the birth of Ishmael. Thus the time of Hammurabi is c. 1850 BC.