Ahmes did not think of pi or irrational numbers as required approximations, searching for a numerical limit, written into rational numbers as we do. Ahmes and all Middle Kingdom scribes (that I know of) used one value for pi, 256/81, based on a rational number standard that is not entirely clear, to anyone that I have read. I suspect the reason for different values of pi were not suggested (by any scribe) is that all scribes had to play by the same mathematical rules, thereby, following the edicts of Pharaoh, and other directives, in this, and other numerical (rational number) respects.
Of course there is the squaring the circle debate. I take the side that Ahmes did square a circle to find his era's approximation for pi, a method that was at least 350 years old, at the time. Yet, formally, no other value for pi has been directly found in any Egyptian texts, written after 2,000 BCE, a fact that Franz has indirected disputed, but has not directly disproved.
Hence, I will not offer an alternative (better estimate, like 22/7) value for pi, as Ahmes may have been capable of offering, as Archimedes surely did offer. Archimedes lived in another culture, one that intellectual readily accepted (and lesser intellectuals debated) that irrationals and higher order numbers were real, and, hence, an open search for a better round off limit was socially and intellectually acceptable.