Alone? No. The transmission of imperial title was a pong game between carolingian dynasty member until it went void, and one could keep the imperial title without being obeyed at all (Charles III is a good exemple, having all his other titles being stripped out)Then could Charles the Bald and Louis the Stammerer keeping the Imperial Crown be a prerequirement to the development of a French HRE?
It would be a matter of keeping the imperial title out of others, undergoing structural changes, and keeping enough power for being hegemonic enough to form an HRE-like.
I think that *HREmperors have better chances not doing that in the same row.
Or was the fact the HRE formed in Germany and the last Caroligian Emperors were from East Francia just a coincidence?
You'll notice that past Charles III, all emperors are chosen outside carolingian dynasty (save Arnulf, but barely as it was probably a bastard) even if having families ties to carolingians.
There common link isn't that they were king of Germany, but kings of Italy : at this time, imperial title was tied up to italian kingship and was given by the pope in order to gain local help in Italy above all other concerns.
Of course, german kings had a better chance on this as they had a direct access to Italy and had less challenging internal situation, when western frankish kings weren't even able to get in southern Gaul by this point.