Not to prolong much longer, but trouble came mostly not from the kings / emperors but from those Great Nobles who, because of the haphazard dismemberment of Lotharingia, were vassals of the French crown for some of their estates but otherwise (practically) independent rulers of their own lands under the HRE umbrella. Specially the Counts of Flanders: while Flandre Royale (West of the Escaut) was part of the kingdom of West Francia since 843 their other domains were 'imperial'. Then their heirs the Dukes of Burgundy, who already were in the same situation with the Duché de Bourgogne French since Richard the Justiciar was made the first Dux Burgundiae ca. 918 by French King Charles le Simple while the Franche Comté de Bourgogne was Franche, 'free', precisely because it was imperial (!) and thus 'free' of French vassalage. They were a constant source of conflict between France and the HRE, even before being supported / used as proxies by the English monarch. The French kings constantly strove to make their realm a pré carré clearly delimited with 'natural borders', *specially between the Alps and the North Sea*.
The French kings constantly strove to expand their power, you mean?
Being vassals of two kings depending on what one held wouldn't magically go away with what OTL was Lotharingia being given in full to France. Unless you make it so that one can't hold lands in France (defined as "as a vassal or subvassal to the French king") and somewhere else, for example.
You could give it to the eastern half with just as much effect.
And of course, this is assuming that the idea of one half getting it all isn't itself grounds for dispute by kings.