Rump Western Roman Empire could be base in France as a Romano-Gaulish state.
You'd need a radically different situation in late Roman Gaul, then.
By the Vth century, the various Gallo-Roman territories still not under the direct control of various Barbarian foedi (which doesn't mean at the latest that they couldn't be under their influence*) were mostly made of aristocratic and urban patchwork, without clear base to support that they were more de facto alliances under the very technical imperial authority (at least since Majorian and Avitus) than states.
Eventually, they were even more easily swallowed up by foedi not only because of their divisions and relatively* lower military capacity, but because their power was based on their aristocratic prestige and you had a lot of their family network and clientele already servicing barbarian kings (such as the Syagrius "Solon", in the Burgondian court)
While Syagrius, son of Aegidius,
recieved a special historiographic treatment which had as prime function to "fill" the map; you had no unified Gallo-Roman entity in the course of the Vth. It balanced between obedience to Roman authority and servicing Barbarian foedi, and with the collapse of the former, there wasn't much viable alternative.
Furthermore, opposing Barbarian foedi and Gallo-Roman structures is a bit moot : Barbarian kingdoms in Gaul ended up being Gallo-Roman states in pretty much most of their institution (with a Babarian influence, itself more or less romanized if not artificial) would it be only because they were acknowledged by Roman authority, integrated Roman institutions but as well integrated the aformentioned Gallo-Roman families (Aviti, Syagrii, Desideri-Salvii, etc.) into the regional and realm management.
For everyone concerned, Childeric and Clovis did represented a valid representent of the Roman imperium in Gaul.
*It seems that Gallo-Roman aristocracy usually had a tendency to support either Goths or Burgondians in the mid Vth
** Various Gallo-Roman or Hispano-Roman nobles ended up servicing military Barbarian kings, hence the "relatively"
The Holy Roman Emperor got the title to fill the vacancy left when the throne in Constantinople was held by a woman
I think you're confusing two things there.
Carolingian dynasty did claimed the imperial title, but not the Roman imperial title as generally understood : in the quasi-totality of Carolingian texts, indeed, they used the titles of "August Emperor"* as they were more interested on claiming rulership over Christiendom as, as you said, it was seemingly broken in Constantinople.
Romanus or Romanorum, however is systematically tied to the people of Rome, and their representend the pontiff.
The Carolingian (and Ottonian up to its last phase) Empire could be considered "of Romans" only in the way that it had been acknowledged by the people of Rome, meaning the pope. Going trough contemporary texts, this is the only explanation and ideological base to the title one could find. Every other narrative about "being crowned roman Emperor" have NO historical base whatsoever.
It doesn't mean that Carolingians (and Ottonians) didn't acknowledged at the latest there was a Roman influence or a translatio imperii, but the matter for them was the use of roman imperial feature (mostly Late Imperial features) was motivated much less as a desire to consider themselves Romans, than to consider themselves kings or emperors in the like of Davidic or Constantinian rulership (as hinted by the decoration of Aachen's palace).
What we used to call HRE (which is to be distinguished from Carolingia, which is as different from HRE than it was from ERE), eventually went much more into the way you describe (especially with Otton II), altough more on a really superficial and non-systematical way than Russia eventually did, as the name "Roman Empire" which at first pretty much followed the line of Carolingian titulature ("August Emperor [of the Empire of the Romans]") for similar reasons, went to its way to rival the Byzantine Empire.
But that's a rather distinct evolution which didn't appeared at first : Charlemagne itself kept using the prestigious titles of "King of Franks and Lombards" way after his imperial coronation (on the golden solidi, for instance, while they were prime prestigious assets) and the imperial title wasn't that supposed to survive his death for what matter the Ordino Imperii of 806.
*You had as well a very short use of the seemingly long and a bit weird title of "August Emperor ruling over the Roman Empire" by Charlemagne.