I can't say much about the Medieval Holy Roman Empire--which is what your concern is primarily focused on. However, by the early modern the various member of the Holy Roman Empire tended to identify much more with their identity as Germans. As you probably now, in 1512, the name was changed to the 'Holy Roman Empire of the German Nation' (Heiliges Römisches Reich Deutscher Nation), both as a recognition that the Italian provinces and the Burgundian lands were 'lost' as well as driving home the importance of the new German institutions, like the Circles, Reichskammergericht, and others. So, it would be fair to say that by the early 16th century the HRE wasn't exactly claiming identity as 'Roman', and they recognised that they were 'Germans'.