Like many ancient cultures Egyptians viewed themselves as the only "true" humans and their country as the only "legitimate" country in the cosmos, this brings about that Egyptians often simply referred to themselves as "humans" (rmT.w, Remetju - .w/u is the plural marker) and to Egypt as "(the) land" (tA, Ta) or "(the) two lands" (tA.wj, Taui), this based on the popular idea that Upper and Lower Egypt (as seperate entities) were united in pre-dynastic times.

The term km.t (Kemet) becomes increasingly popular during the Middle and especially in the New Kingdom and has often been described as the somewhat "official" denomination of Egypt. km.t is derived from the stative verb km(m) (kem(em)), "to be black", and is as such the famous description of the fertile, alluvial soil of the Nile with its distinctive dark colouring, figuratively meaning something like "the black land". km.t could be used as a collective noun, then meant to describe the entire population of Egypt. So, "Kemet" may either mean the country or the people of Egypt (the difference between both uses is appearant through context and in script, km.t as a collective noun uses different determinatives). Another possibilty is the use of a nisbe - nisbes are adjectives that depend on a noun (km.t in this case), which can be nominalised in return. Nisbes are formed with a suffix .j - so km.tj (Kemti) would be "Egyptian", km.tjw (Kemtiu) would be "Egyptians". The .j is seldom written though, so the distinction between them is not always as easy as it appears to be here, in an isolated context. Less complicated and used quite often is a simple genitive construction - rmT.w nj km.t (Remtju ni Kemet), "people of Egypt". That expression actually lived on into Christian times and is the mainly used form in Coptic - "rm-n-kēme" (Southern dialects, pronounced /Rəm-ən-Kēme/) or "rem-n-khēmi" (Northern dialect, pronounced /Rem-ən-Khēmi/ ), which literally means "Man of Egypt" (Coptic uses the old Egyptian .w plural very rarely, instead a simple plural article is sufficient - "(the) people of Egypt" would be "n-rm-n-kēme"); "rm" is a vowelless form of "rōme" ("man, human", the Ancient Egyptian rmT) and as such functions as a nominal prefix, "n" is the genitive preposition (the Ancient Egyptian nj) and "kēme" is the Ancient Egyptian term km.t. Your "rem-en-kimi" seems to be a weirdly vocalised version of that, but since Coptic actually denotes vowels (in contrast to Ancient Egyptian) there are none to be pronounced where there are none written, so that's just unecessary.

Your discovery of "Kememu" seems to stem from Budges rather infamous "Egyptian Hieroglyphic Dictionary", where that is listed as "Kammau", I would refrain from using that term. I can't look up his references right now, but it's obvious none the less that he misread parts of the Hieroglyphics (e.g. he actually includes the plural .w whereas that is clearly meant as a determinative in this case (since it's written after the other determinatives) and shouldn't be read). I can't find any other mention of this, neither in the great Wörterbuch der Ägyptischen Sprache nor in the Thesaurus linguae Aegyptiae and I haven't ever come across it myself.

Similar to km.t in its use is tA-mrj (Ta-meri), which means "beloved land" and is of course another name for Egypt. Like km.t tA-mrj could be used as a collective noun that denotes the entirety of Egypt's population. As with "Kemet" "Tameri" may describe both, land or people; Tamereans may be an anglicised form of that, though I personally never have encountered any mention of "Tamereans" (it makes sense though as Tameri is not real plural and may be confusing to look at, considering it either denotes the people or the land, whereas "Tamereans" are clearly people).

The problem with all of these terms is that very few people will actually understand them, since - even in an academical context - it's highly unusually to speak of the "Kemetiu" or the "Remnkeme" instead of "Egyptians"; I have yet to witness that, to be honest. Very few people have access to Ancient Egyptian dictionaries or any skills regarding those languages whatsoever - and while the use "Kemet" is certainly quite common as a denomination for the ancient country itself, Egyptian circumscriptions for the people of said country are not. If you're looking for some variation it's probably better (and easier) to try and be creative with your own language - call them "Children of the Nile" or translate the terms given here literally - "People of the black land/soil", "People of the beloved land". If you're working in an academical context you could use the Egyptian transliterations (like km.tjw) or the Coptic script for Coptic words (ⲛⲣⲙⲛⲕⲏⲙⲉ instead of "n-rm-n-kēme"), that would certainly be less awkward than using some anglicised transcription.