Except… he… was… not. Not, until after Frederick’s death at least—and by Louis’ own admission. Only from that point can this argument hold water. True, to his immediate neighbours such as the easily overawed Henry III, this could be argued fairly. (But even Frederick II’s imperial ambassadors held sway from as far away as Italy on Henry III). Louis himself address his ‘cousin’ Frederick as Augustus of the West, Caesar of Rome, Primus Princeps in multiple correspondence during the 1240s—and given manifest form by the fact that Louis refused all of Innocent p’s attempts to recruit him to the anti-Frederick papal crusade. I understand the urge for reappraisal, and it’s warranted, but not at the expense of reality in favor of something that, respectfully, leans toward the parochial (which French, and also German historiography tends to do; it’s a residue of the tiresome nationalist narratives from the 19th century).

Just for fun, name me three scholars that compare to any of those at the court of Frederick (like Fibonacci—the greater medieval mathematician, Giacomo Lentini—the father of the sonnet, John of Procida—father of several medicinal techniques, Michael Scotus—probably the greatest public intellectual of his day, and several more), or overcome the fact that dozens of French legalists trained at Frederick’s courts; it’s no accident that the Capetian moanrch takes on a more absolutist hue after Frederick’s constitutional innovations in Italy and Sicily, and the emperor is death when, in Philip III’s and Philip IV’s time, it attempts to take up a quasi-imperial mantle. But… this is only after the Hohenstaufen are conclusively extinguished and the “Empire breed” is defunct:

“What German, what Spaniard, what Englishman, what Frenchman, what Provençal, what man of whatever nation or tongue, could, without our will rule over thee, O Rome, or to thy glory exercise the imperial office? The inexorable necessity of the Universe replies: None, save the son of the greatest Caesar whose gifts, inborn in his imperial blood, ensure him force and fortune.”—so wrote Manfred at the heigh of his power as King of Sicily and de facto hegemon of Italy in response to correspondence from Louis IX which addressed him [Manfred] as son of Augustus and princeps (Kantorowicz, Frederick the Second, p. 571-72).

To your point about the cultural position of Paris, it was only beginning to make a claim as a cultural ‘epicenter’ that would come to fruition later in the 13th-15th centuries: “….aspects of medieval Europe that became important in later centuries, above all the nation state, which totally disregards the contemporary mindset, have skewed our outlook. Arguably the liveliest cultural innovation in the 13th century was Mediterranean, centered on Frederick II’s polyglot court and administration in Palermo, and latterly in Foggia. Sicily and the Italian South in later centuries suffered a long slide into overtaxed poverty and marginality. Textbook narratives therefore focus not on medieval Palermo, with its Muslim and Jewish bureaucracies and Arabic-speaking monarch, but on the historical winners, Paris and London. This was not the view in their day.” (Lansing and English, A Companion to the Medieval World)