What I wonder is how much more enticing to convert this new Christianity would be, or if it is going to be detrimental to its expansion. Will it lead to barbarians not adopting Christianity, or at least not adopting Donatism? After all, the last thing various post-Western Roman states want is the Emperor in Constantinople having a major say in their affairs. Therefore, I can imagine Donatism being eventually restricted to the Eastern Empire and the territories it influences, if only for political reasons.
The Germanic invaders all (except for the Franks) embraced the Arian heresy before invading Roman territory. The Franks converted from paganism to Nicean Christianity and the Visigoths eventually converted to Nicean from Arian, but the Ostrogoths and Vandals both clung to their Arianism until their tribes were wiped out during Justinian's reconquest. The common people stayed Nicean during this period, which is one of the reasons that their was so much conflict between the Germanic invaders and their Roman subjects. So the Emperors in Constantinople would be exercising control over the subject peoples' not over their Germanic monarchs.
The Inquisition would probably provide an important lever for the Emperors in Constantinople, especially in the event of something like OTL Justinian's attempt at reconquest. The western church would already be under the direct control of the Emperor (since the Emperor would control of *Inquisition) and so the western church could probably be used to much greater effect in the process of the conquest.
This will really ripple history, but I suspect the Franks would convert to Nicean Christianity eventually, since it is politically the smart thing to do (allows them to connect to their subject people, get more support for wars against the heretic Germanic rulers of (Visigothic) Spain and (later, Lombard) Italy. The Franks had a good relationship with the Empire for quite a while, and I think Justinian officially recognized the Frankish Kingdom as legal, something that he didn't do with the other, heretical, Germanic rulers.
Anyway, I think the western Kings would be able to exercise control over the Inquisition in their own country. The eventual split with the East over Latin and control will probably occur dispite the switch to Donatism. I can see the evolution of an independent Church in the West (as per OTL) simply because of the chaotic nature of the Germanic tribes politics. The Church was able to maintain continuity in a way that the Germanic tribes simply couldn't, which allowed the Latin Church to become so independent of the Germanic Kings.