Χαμ in the headword phrase is clearly to be taken as genitive based on the context. However, like other Greek borrowings from Semitic languages, the word does not receive inflectional endings.
[1] Otherwise Μεσραίμ , usually transliterated from the Hebrew מִצְרַיִם as "Mizraim" in modern English versions. See Genesis 10:6 and I Chronicles 1:8. See also mu 674 (Μεσρέμ ).
[2] Theodoret (PG 80.1496c) on Psalm 77.51 LXX.
[3] In modern Arabic, the word for Egypt is ﺭﺻﻣ (MiSr).