Interesting new chapter. I appreciate that you're taking the time to really delve into Church topics - too often many timelines focuses primarily upon the political and treat religion as an afterthought. ANd don't worry about the irregular chapters: I really need to get back to my own timeline after a year off, so you're far more regular than I am. real life happens, after all!
Since an independent Carolingian Aquitanian Kingdom didn't exist in our timeline after Charles the Bald's transformation of the title of King of Aquitaine to a title in name only, I personally find it to be necessary to take a closer look at the structure of the kingdom and all the butterflies it entails. The Cluny Abbey of OTL, for example, was most definitely shaped by its social and political environment within West Francia, which is not as similar to TTL's setting as it might appear on the surface, especially after this world's formation of the HRE under Hugh I. Combined with some of the more subtle differences of TTL's saeculum obscurum, it will lead to a different Church of Rome by the next century, so I needed to set the stage for that by explaining at least the most important parts of the contemporary ecclesiastical life ITTL.
As for the irregularity, absolutely agreed. With the outside world slowly but steadily resuming to a new normal, the time for writing and researching has been sometimes lacking on my end, regrettably. But oh well, stuff breaks, life goes on.
The political differences between the two branches of Church reform will definitely prevent the Church from acquiring so much power vis-a-vis secular rulers -- now they can rely on Florian monasteries as a ready counterbalance to the idea of a Church hierarchy independent of temporal power. I'm sure the Mechlinian reforms will find more purchase in more decentralized realms, but its hard for the Papacy to counteract the strength of kings with their support and the power of their monasteries.
I hinted at Hugh I and his ominous successor not feeling completely confident in the Florian monastic reforms as well as other streams emerging in abbeys such as Farfa or Fleury, so you can safely expect more such branching reform attempts, similar to OTL with the abbeys of Gorze or Hirsau giving rise to different attempts at reforming the sorry state of monasticism at that time and, over some corners, the role of the Church in the Occident. You're absolutely right in that these reform attempts are also of political nature due to the state of monasticism in medieval Europe at that time, be it ITTL or IOTL. So, one kingdom might treat these issues differently as others
, as mentioned with King Wipert I of Neustria. He will surely not be the last king of Europe to look for alternatives to the Florian Principles which do allow some level of lay, and thus potentially rival, influence. Nevertheless, it will remain interesting how the Papacy will develop, given their political and ecclesiastical framework of TTL, though I can already assure you that it will take a different trajectory than IOTL.