You need a less successful Otto I for this to happen, without hurting Germany too much in the process. OTL he greatly reduced the autonomy and power of the different dukes, making them firmly his subjects instead of equals. He then went on to defeat the Hungarian nomads at Lechfeld, ending their raids in western Europe. Which boosted his prestige in Christian Europe, being viewed as the man who defeated the pagans. He then proceeded to conquered Italy, and was crowned Emperor in a similar manner to Charlemagne.
So all in all who he was a very competent man. For Western Francia to be able to claim the imperial title I imagine they must at least get their hands on Italy, at the very least. I believe having Otto I still defeat the Hungarians, but die before he invades Italy would be needed as well, or have him refrain from doing so all together. This would leave his son with relative firm control of Germany, without being overextended and squabbling with the pope in Italy and Muslims in Sicily. The problem with Otto I dying too early though is that is son is too young, Otto II was an infant when his father won the battle of Lechfeld and he was 6 years old when his father conquered Italy.
I agree, that this was very imaginable.
For this (a French HRE) to happen West Francia needs to control Italy (and thus have influence over the Papacy) and for that they need to control Burgundy (to control and have easy access to the Italian peninsula). Lotharingia OTOH could go either way, but with West Francia having to focus on Italy and Burgundy, I can still see East Francia prevailing there (if only because West Francia needs to focus elsewhere). Furthermore IOTL East Francia didn't need Burgundy to have direct access to Italy, whereas ITTL West Francia needs Burgundy.
Agreed, this French HRE need to be bigger than the OTL Kingdom of France in order to create several rival centers of powers. Ile-de-France, the Lowlands, Burgundy, Aquitaine and the Lombard region would be a good start. Could probably give them all of Lotharingia, albeit not necessary, could just go with the Rhine border as suggested and give Holland and Friesland to Germany. Keep them busy with the Pope, internal power struggles and conflicts with/in England, Italy, Iberia and of course Germany and I can see such an HRE being very interesting to follow. Especially if events like the Crusades and the settlements of the western hemisphere still happens at some point. I am very interested in seeing how it would play out of this HRE becomes as decentralized as the OTL HRE was. Would settlements made by individual princes still count as part of the HRE? The potential for crazy scenarios is pretty high.
That leaves a German Kingdom that can look several different ways depending on when the split happens. If it happens around the time Otto was made HRE OTL, a few years give or take before 1000AD then the German Kingdom (assuming a Western/Southern HRE like suggested above) would have no major territory west of the Rhine, the duchies of Sachsen, Franken, Schwaben and Bayern would be the core territory of such a state, and the marches in Austria and OTL Brandenburg/Mecklenburg/Saxony would already have been established. Then the question becomes what happens to Lotharingia, Bohemia, Moravia and Verona. I can see the French HRE getting all of Lotharingia, pretty much a border at the Rhine, which ruler gets Holland and Friesland is up for bets.
Verona is sketchy as well, at this point it is a march and was held by the Duke of Bayern, so I assume it would initially go to Germany. But I can imagine it being contested by the French Emperor, with it being right next to Lombardia. That said, it's fairly far away from the the core "French" territories, an Emperor focusing his power so far east might face trouble in the west. The same issue the Hohenstaufens had with Southern Italy vs Germany, except now it will be even further spread out, going from Antwerp to Paris to Bordeaux to Milan to Rome.
The last question then is what becomes of the Duchies of Bohemia and Moravia, I assume they remain a part of Germany. But it will be interesting to see how it would work out ITTL, for one it would never be granted the standing of King by a German King. OTL it was made a Kingdom by 1198 and recognized shortly after in exchange for taking side of elected Emperor Philip of Swabia against his rival for leadership Otto IV. See no reason why it would be less integrated than OTL.