Abraham was close friends with an Amorite, Mamre, and his brothers. The Amorites were somewhat Indianised in culture (as were several other peoples). Sarai’s name is related to that of the Sarasvati River which had been progressively drying up over the previous 1,000 years and had been as important as the Indus and the Ganges, which fact had no doubt helped motivate emigration from that region.
The campaign related in Gen 14 is typical of a “coalition” of the time which were trying to keep the caravan routes open. Not all the troops may have been led by a major king or emperor in person but by a general in their name. Some commentators have said that Ellasar is Larsa, but I rather like Father John McKenzie’s choice of Ilansura near Harran where a branch of Abraham’s family were - frictions with an old neighbour! (The ‘n’ dropping out is plausible linguistically.) (On the other hand Ari-aku is said to be the son of Durma-Ilani - Durma being related to “dharma”.)
Amraphel may be an alternative form of a common name of the king concerned: I can’t remember where I read it contained the name Sin (moon god) who was also worshipped in Harran (they used a moon-based calendar). Chedor Laomer (Laghumal) may be a colloquial variant on Chedor Nahhunte according to McKenzie and he was active some time after 1650 BC, as was one of the kings called Tudhalias (Tidal) of the Hittites. Though some people place Abraham a couple of hundred years further back.
It’s not enough to say he didn’t exist when we don’t have scrolls, tablets and such in glass cases with his signature or entry in a birth register. Even quite important people didn’t write in those days and in any case not many documents have survived from that period.
The period of uncertainty in his dates of living - a few hundred years - are nothing compared with the Indians Krishna, Rama and many others where the range is several thousands of years.