This is the thing, really. The point of the popes was they wielded a massive amount of worldly power in medieval Europe. Your typical D&D religion is based around personal polytheism. An individual cleric might wield lots of magical power, but that doesn't typically translate into political power.How many places have one head of the church? Do all the gods have a head of church? Do all of them get together once a year to create rules for the little people? Can a King just declare the Pope is no longer in charge of the church and place himself there?
5. A deity evil or insane enough to want their followers to fight each other.I think you'd need a few things to get a decent "pope war" going in D&D.
- Distant/absent gods who do not directly respond to divination magic, but use angelic intermediaries who may or may not have direct contact with the gods themselves.
- A religion that's organized enough to have at least a regional if not worldwide structure, with centralized control. No pope, no pope war.
- This religion needs to be dominant enough that its followers make up a large portion of the population.
- The religion also needs to wield enough worldly power to turn it into an actual war instead of just strongly worded letters – either directly through its own forces or indirectly through influencing those who do have armies.
I recall that Power of Faerun had a chapter on religious authority that talked about this a lot, albeit in the context of religious schisms. It was some pretty good stuff.
A variant of this would be Terry Pratchett's Small Gods. The religion has become so dominated by doctrine and structure that what people really believe in at this point is not the god, but the Church. Almost nobody has a direct connection to the deity anymore, so their voice and message has been lost.I'm doing something similar. One scenario is real gods incapacitated two churches are dueling it out. Fake gods leeching gods power granting spells in their name. Incapacitated god can barely grant spells.
Other one is real gods been weakened enough his high priests are heretics. Either hacked by different power or in 5E you get spells anyway.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.