Originally, I was going to trot out the old saw about the HRE being neither holy, nor Roman, nor an empire, but I thought it was too cliched. With the debate of whether the HRE should be in the game at all, that old quote is actually relevant.
Anyway, as regards the flavor units, what if the early units (Archer, Axeman, Spearman, Swordsman, Horseman) were Frankish, and the later units were Renaissance Germanic?
Someone has uploaded a Landsknecht arquebusier to the DB, and I still stand by my earlier suggestion of putting the HRE eagle on the German flavor units. That takes care of the Medieval/Renaissance units.
I was hoping to find an illustration of a Frankish axeman in my Osprey "Age of Charlemagne" book, but they did not choose to include one. What I had in mind was the early (pre-Carolingian) Franks, who attacked in a single deep infantry column, and who hurled the francisca (a type of throwing axe) and javelins at the enemy before pressing home the attack. They were unarmored. This was about the 6th century CE, so the Franks were still ruled by the Merovingians, and not the Carolingians. (If anyone had a claim to the Western Roman empire at the time, it would be Justinian and Byzantium, although the Gothic Kingdom of Italy had the territory.)