For the purposes of this question, let's say Charlemange was not excited by the prospect of being crowned titular emperor of a dead, non-frankish empire
Which summarize roughly the IOTL situation, interestingly. Carolingian Empire wasn't considered as the resurrection of Roman Empire (in spite of its popularity, it have zero credibility when one look to sources, and is generally academically discarded)
Calling Carolingian as "Roman Empire" or "HRE" is at best making an anachronism and not understanding at all the nature of royal power and institutions in the IXth century; at worst deliberatly ignoring sources in favour of biased historiography.
(From the one pulling that in first place, I understand you didn't created this notion just for the thread, of course)
and decides to have the Pope crown a male from one of the few remaining Roman aristocratic lines.
The main problem about this, particularily, were that Roman aristocrats were close , maybe a bit too close, to ERE.
One shouldn't confuse pontifical and monastic institutions (that indeed knew a growing temporal power over the Ducatus Romanus), and Roman families.
Eusthatius, the last byzantine duke of Rome before Carolingian takeover (you had other dukes up to the IXth century, but they're growingly mixed with pontifical offices) had enough ties with ERE to be sent as an ambassador on Ravenna.
You can have another exemple of this defiance with the events of the Procession of the Greater Litanies in 799, where Romans or at least powerful factions, attempted to get rid of Leo III.
Or the factions fighting on pontifical institutions, more or less related to religious disputes with Constantinople (as Syrians) but also political.
Charlemange of course retains all the power as king, with the potential HRE acting as a puppet figure head for him to control.
There's the greater problem about legitimacy. Virtually all reference to "Roman" in Carolingian texts, is about the pontiff (or the
Romanus populus as in the inhabitants of Rome and their "natural leader" the pope).
The idea of Francia was a Christian dominion, having the dominion over Christians (in the similar idea that Constantinople had such and subsequently lost it, at least on legitimacy grounds) was extremely tied up with the Roman question.
Charlemagne didn't wait for 800 to meddle with religious matters, as can be pointed with the Council of Frankfurt, and renouncing the inter-dependent relationship between the pontiff and the king would have represented a huge problem of legitimacy and royal ideology : don't forget that this relationship began to form since the early VIIIth century at this point and that it coincided with the legitimisation of Peppinid/Carolingian takeover.
Crowining someone else as an Emperor, evenmore so a Roman, would have meant toppling all of the institutional/political/royal edifice built by his predecessors for no really good reasons.
As an aside, I'll add that a good deal of Carolingian justifications of their rise was they had to topple feeble kings that were mere puppets (of course, there's a huge part of what we'd call propaganda there). Just creating a similar situation for no good reason makes really little sense when it comes to contemporary mindset.
So, to be honest, it's quite absurd.
Now, if you look at the IOTL situation : we have a papacy that was an important identitarian and political marker, and that had an important Syrian or Greek presence up to the mid VIIIth century.
Charlemagne and his successors managed to "exploit" a localism that existed in Rome and that favoured Roman popes, to help cut off links between ERE and papacy even more than they were.
You had to wait more than one century to have popes that weren't Roman in origin, acting when it comes to Carolingians, while more than a mere figurehead, as a dependent institution.
A large part of what you describe in your OP did happened, for the papacy.