The problem with Middle Francia was its total lack of cohesion and its neighbours.
While WFrancia beneficied from a relatively good coherance ("nationally" speaking, being most of the former core of Francia) and that EFrancia benefited from such (stem-duchies forming a relativly good cohesion itself), and being not onlr rich but with expansion features, MFrancia was right in the middle, no better than a buffer zone.
Geopolitically, it was doomed to be subject to its neighbours conquest or at best control (as for being the deciders on which part goes to who, king or son of a king)
You'll need a catastrophic decline of both Western and Eastern Francia, that would miraculously spare Middle-Francia to get rid of that.
Even if Middle Francia avoid divisions and benevolent protection from other Carolingian kingdoms, it doesn't start very well : its territories weren't the most rich, or when they were, particularly prone to raids (Vikings and Saracenic) that the quite weird form of MFrancia prevents to really crush efficiently.
Admitting that the kingdom keeps its integrity, I'd tend to think MFrancian kings would be too weak to really claim if not build the HRE out of nowhere.
The only way I could see MFrancian kings to do so, would be to be chosen by either WFrancians or EFrancians as their kings at some point (but that's unlikely, as they were too weak to be usually considered as possible candidates) and with their common kingship (Italy being required for claiming the imperial title) build a new imperial institution.
But it wouldn't be exactly a MFrancian becoming such, and giving the premises, it's highly implausible.