What if Lord of the Rings was re-enacted in early 13th century Europe?

I've been exploring this idea for a little bit. Some of the connections are obvious: Byzantine Empire as Gondor and the Fellowship is the fourth crusade. And, instead of what the crusaders did in otl, they would have to bring an emperor with super strong roman/byzantine lineage whom the empire accepts (analog to Aragorn, or course).

Beyond that, the connections are a little more tenuous: a pope who is also a general as an analog to Gandalf? Vikings as elves? British or Irish monks as the hobbits? A treasonous patriarch as Saruman?

I really DON'T like the way religion would play into this, but in a full re-enactment, it would have to. To that extent, the Ayyubids and sultanate of Rum would be the Enemy, and they would perhaps some rulers would be converted to Christianity? The Catholocism/Orthodox Christianity divide somehow feels harder; perhaps the best solution would be some sort of merging of the churches, in a way that both would accept? It seems like accepting the catholic church as part of the orthodox church might be the best way...

If we are to take this re-enactment all the way, the Grey Havens part is even weirder; seemingly the best way to re-enact it would for the pope, along with some vikings and northwestern Europeans to go to Greenland and perhaps those North American settlements (the latter of which had been abandoned around 150 years before this)