Steve, you're right with the adoption of the double headed eagle from Byzantium. My guess: the double eagle was adopted when a Byzantine princess was married to a German king. Using a coat of arms in a composite one is to show ownership or a claim thereto.
My native town was the first community which sealed with the double eagle in the 14th century.
Mudguard, the double eagle is not a supporter of a shield, but it is the main part of the coat of arms. To the Imperial Eagle of Germany the Habsburg family added their own shield (red-silver-red) on its heart to show who reigns the Empire. My native town added its colours silver-black to the Imperial eagle to show that it was an immediate Imperial town, subject only to the Emperor with Imperial privileges.
First German king and emperor of the house of Habsburg was Rudolf (1273-1347),Albrecht I (1298-1308), Frederick the Beautiful of Austria (1314-1330), then the house of Habsburg reigned the Holy Roman Empire from 1438 to 1740 and from 1745 to its end in 1806.
Let me put stress on the note that the double eagle has nothing to do with Austria and Hungaria, but that is a symbol much older.

Wilfried