Yes, a strong empire could easily manipulate French politics by playing the Feudal lords and the king against each other, not to mention playing hte spicy English ingredient.
Yeah, with a centralized HRE it becomes much easier that the English and the HRE can team up to wreck the rise of a centralized French rival. It could end up in a partition, West to England and East to the HRE (and perhaps parts of the South to Aragon or Castille), or France could become a disunited buffer area and contested playground between Britain, Spain, and the HRE.
However I think a centralized HRE would not include Italy (and Bohemia would more or less be a puppet state).
Bah. If HRE centralization happens in the X-XIII centuries, where the most feasible PoDs cluster, both Italy (including southern Italy), Low Countries, Switzerland, and Bohemia are most likely going to end up integral parts of the empire, and almost surely so if France is crippled by decentralization. Back then, there was no significant nationalism, and if the emperors are able to crush particularist opposition in Germany, so they shall in Italy and Bohemia, there was no meaningful difference back then. The empire can easily use the "second Rome" idea as a unifying ideology, which in modern times becomes multicultural HRE nationalism.