Hmmm, tough one. Republican government, even of the limited type described here, is different enough from anything else the Egyptians had that they probably would develop a distinct word for it. Given that a kenbet is at root a law-court, they might refer to a republic as "province of the judges" or something similar, much like the compilers of the Old Testament referred to the legendary pre-kingdom days of the Israelites. This might hold true even in republics that the kenbut do not lead or lead only indirectly, because the kenbet-state was the original model.
There might also be a term for the new monarchial philosophy that the dynasty in Henen-nesut is pioneering - "just kingship" or something equivalent. Divine justice is central to Egyptian political ideas of TTL, even more so than in OTL, so their political terminology would probably emphasize justice, judges and law.
Of course, I'm willing to listen to any better ideas.
This question - what a native Egyptian word for "republic" would be - has been bouncing around my head since the start of this TL. So far, the best I think I've come up with is makenbet ("maqenbet" as an alternate spelling), which means "that (thing/land) which is under a kenbet). "Ma-" in Middle Egyptian is the so-called "m- preformative prefix" which, when attached to a root word, indicates a place where/instrument with which something is done.
"Just Kingship" can be translated literally as sutenyet-ma'at (or "nesutyet-ma'at" - there's disagreement over how the first word should be transliterated/transcribed).