I could certainly see this in a form - a West Francia HRE would likely, at least in the short term, have control and influence over Italy, the Low Countries, Burgundy, and maybe Spain and the Rhineland would likely be contested between them and East Francia. In the long term, it would start losing them like Germany did.
The East Francia region may solidify into a number of states that will eventually form Germany. Whether it is Saxony, Bavaria, Austria or someone else that forms it is the question.
With a more centralized France during the Viking Age we may see more of the Norse focus on Britian and Ireland, and we already butterflied away Normandy and William of Normandy with it. We may see Canute and his kin form a unified Scandinavia, one with a firm grip over a far more Nordic British isles.
With such a strong nation on it's border, Christian Spain will have a much more serious morale boost, and may get a few crusades directed toward al-Andalus.
I agree, that a West Francia HRE would have to control Burgundy and Italy, but Lotharingia (northern Middle Francia

) could go either way, especially with West Francia having to focus on Burgundy and Italy.
Instead of turning east this West Francia HRE as the leading Catholic monarchs might prefer crusading in Hispania instead.
However West Francia becoming and staying centralized will be tricky, though not impossible; the OTL HRE under the Ottonians was further on that road than their OTL French counterpart, but eventually things turned out differently.
OTOH I disagree about East Francia, I don't see it disintegrating into a number of states. Sure it was an elective monarchy with a few strong stem duchies and it will evolve into a German Kingdom. In fact, if a dynasty, whether Carolingian or Ottonian (maybe Salian), manages to survive long enough, then like IOTL the Capetians managed to do in France*, the elective monarchy might develop into a hereditary monarchy.
(*= IIRC France was formally elective until Philip August).
I can even see East Francia managing to gain Lotharingia, since like I wrote earlier, West Francia will be preoccupied in Burgundy and especially Italy. The situation in Central Europe might be a bit different too, East Francia, later the German Kingdom, won't have to focus on Italy, so I can see them being more successful there.
For instance IOTL East Francia/the German kingdom, temporarily lost parts of their (northern) East marches after a Wendic uprising, in part, because they were preoccupied in Italy; in fact the Imperial army had suffered a defeat in Italy. ITTL a ruler of East Francia could send his entire army to crush the rebelling Wends.
A East Francia/German kingdom could still control the same territory north of the Alps the HRE did IOTL 1000 AD. However the kingdom of Burgundy (either the HRE & king of West Francia is their king or they have their own vassal king) will be a part of this West Francian HRE, and the HRE as king of Italy will also (nominally) control the marquisate of Tuscany, Lombardy, the march of Verona, the march of Istria, Pentapolis, the duchy of Spoleto and (disputed by the Papacy) the Papal states (the latter as HRE).
BTW the control of Burgundy is a practical one, since in order to be able to have influence in Italy and directly on the Papacy, this West Francian HRE will have to control mountain passes leading into Italy.
https://upload.wikimedia.org/wikipedia/commons/f/ff/Holy_Roman_Empire_1000_map-en.svg?uselang=de