Essentially, I've always loved the idea that instead of causing the Schism, that the lasting legacy of the Pope in Rome would be to lead the way in introducing the western Europeans into the church, and through the church, the Empire itself. Choose whatever motivations you wish, but the end result being that the Pope wants more Exarchs serving the Emperor.
So in this vein, could the Pope foster (successfully) a powerful hybrid of the Imperial Cult with Christianity. Its central tenant is that the Emperor as the Vice Gerent of Christ on Earth, and as such serving him is serving Christ. The order being focused on expanding the Emperors realm against the Heathen, with another side of it being that the Order may even be used to depose Exarchs, or an Emperor seen as false.
Adding in the belief that the title Exarch is an incredible holy blessing, whereas the Emperor is in reality granting the title to those that are most useful (pious vs powerful).
Could a Western Europe partially gripped by such a cult bring both western Europe under the imperial banner, but push back against the Muslim world? What other side effects could there be?