The Nazis' "obsession" with resurrecting an ancient German past was more apparent than real. The NSDAP adapted historic symbols very selectively with an eye to making these symbols fit its current needs. There really was not much to be gained by the resurrection of the HRE, either title or institutions, for the NSDAP. Hitler and his movement often styled themselves as the culmination of the forces of German history as well as a modern party with an ethos that suited the industrial age. This contradiction, termed memorably by Jeffrey Herf as reactionary modernism, created a somewhat freighted relationship to the German past. The Nazis would mine the nation's history for heroes and validation, but it would simultaneously claim that the movement was fundamentally new.
The Nazi ensign symbolized this dichotomous relationship to the past; the black-white-red colors harkened back to the flag of Imperial Germany which had been replaced by the Republican black-red-gold flag, but the NSDAP ensign used a completely new symbology (swastika) and composition to create a flag that was both new and familiar. The dictatorship initially resurrected the old imperial tricolor as a national flag coeval with the swastika ensign, partly as a sop to old-line conservatives like von Hindenburg, but eventually abandoned the old flag in favor of the party banner.
Likewise, the NSDAP's relationship to its imperial forebears had a similar pattern. Building on völkisch theorists like Arthur Moeller van den Bruck, who first popularized the term Third Reich in 1923, the Nazi government initially styled itself as the third German realm following in succession from the First Reich (roughly from Charlemagne to 1806) and the Second Reich of Bismarck and the Hohenzollerns. In a pattern familiar to other aspects of Hitler's rule, the dictator soon reversed himself in June 1939, forbidding the term because it implied that there could be a Fourth Reich. Fittingly, there is evidence that Hitler started using the term again towards the end of the war and despite the ban and its non-use by the government- the state used Deutsches Reich and after 1943 Großdeutsches Reich in official documents - the moniker Third Reich stuck for the German government between 1933-45 even if the dictatorship did not wholly claim the term.
There simply was no real impulse to revive the HRE or imperial titles in this context. The Nazis' approach to the German past was always functional. While the Nazis tended to celebrate Hohenzollern monarchs like Frederick the Great, they did not have much to say about Wilhelm II. The Nazis actively courted the old imperial nobility, including Wilhelm II's heirs. But this courting always had a subtext that the old elite had to play ball within the new structure of the dictatorship.
The HRE carried with it certain baggage that had unsavory associations for the Third Reich. The monarchy was elective and, even more damaging in the eyes of National Socialist ideologues, closely associated with the Papacy and Christianity. Moreover, the HRE's long and storied history meant it incorporated non-Germans into its polity and much of its later history (ca. 1618-1806) was characterized by weakness, foreign exploitation, and internal squabbling. Himmler would in numerous speeches in the early thirties castigate Charlemagne for bringing Christianity to the Germans, depriving them of their pagan Teutonic heritage, and castigated Louis the Pious for allowing Jews into the realm.
The closest the Third Reich got to a cult of an HRE figure (and he was only an HRE member only in the loosest sense of the term) was the SS-sponsored celebration of the Ottonian King Heinrich I. On the millennial anniversary of Heinrich I's death in 1936, Himmler inaugurated King Heinrich Day. In a speech, Himmler emphasized that Heinrich I showed a true commitment to the German character by refusing to be crowned by the Pope. As the SS head put it, Heinrich I:
reintroduced the old and yet ever new Germanic principle of the loyalty of the duke to his liegeman, in sharpest contrast to the Carolingian methods of ruling based on church and Christianity.
Thus not only was Heinrich a forerunner of the anti-Christian principles of the SS, but his methods of rule prefigured Führerprinzip. Although some historians hold that Himmler thought he was the reincarnation of Heinrich I, Peter Longerich maintains it was much more likely that the target of the SS's celebrations of this Ottonian monarch was Hitler. SS publications repeatedly stressed the parallels between Heinrich I and Hitler. Naturally, Himmler's celebration of a feudal order of enlightened leader and his knightly paladins mirrored his own conceptions of the SS's position within the state hierarchy of the Third Reich. The neo-feudalism of the SS, which selectively mined archeology, folklore, history, and invention to create a historical patrimony for the organization as a reincarnation of the Teutonic Knights, the "Germanic" period of the HRE, and a pagan Germany all rolled into one.
Himmler's interest in the medieval period though was not shared by others within the Third Reich. Goebbels's diary referred to King Heinrich Day with a degree of sarcasm and the neo-feudalism of the SS was the subject of a degree of jokes at the Reichsführer's expense. Hitler himself had relatively little interest in the occult, neo-paganism, or recovering a lost German past. Instead, the dictator tended to approach co-option of the past very functionally. According to Speer, Hitler often mocked the idea that Himmler would preserve his body like a medieval saint. In a 1935 NSDAP conference, Hitler would praise Charlemagne not as a French conqueror of the Germans, but as a German Emperor who like others:
who with a merciless sword and disregarding the fates of the individual tribes strove to bring more German people together.
Hitler later forced Himmler to silently recant his criticism of Charlemagne and have the SS celebrate him as one of the founders of Germany's position in Central Europe. Such an opinion reflected a more mainstream interpretation of Charlemagne that claimed him as a German figure from the distant past. Likewise, although Hitler shared the anti-Christian precepts of Himmler, he was not willing to court an open breach with the churches by direct state sponsorship of neo-paganism.
So in an improbable scenario in which Hitler declared himself Emperor, it would have likely followed the channels of memory dug by Himmler and his ahistorical mishmash of neo-feudalism and neo-paganism. Hitler as Holy Roman Emperor would have fit Voltaire's famous aspersion "neither holy, nor Roman, nor an empire" far better than the HRE ever did: it would have been explicitly pagan, definitely not Roman, and only qualify as an empire.
Sources
Dennis, David B. Inhumanities: Nazi Interpretations of Western Culture. Cambridge: Cambridge University Press, 2012.
Longerich, Peter. Heinrich Himmler. Oxford [England]: Oxford University Press, 2012.