The only option that I could think of is using the EU3 Curia system before HTTT overhauled it.
In that system, there's a list of cardinals, and you can influence them to support your country. You influence them by essentially bribing them. If you have the most cardinals, then you take over the Papacy. (There's also a cooldown period after each successful bribe, so a cardinal you bribed doesn't immediately get bribed again...this ensure some stability within the Curia.)
Now, sometimes, old cardinals can retire and new cardinals get generated based on territorial control (so if you control 25% of the Catholic world, you have a 25% chance of a cardinal spawning that automatically supports your country). But these new cardinals can be influenced fairly easily, again through bribes. If there's a tie, then the previous controller keeps power.
Now this picture doesn't show France taking over the Papacy even though it controls the most cardinals. That's because France is excommunicated, so their cardinals don't count. Barbant was also excommunicated, so Sweden (with its lone cardinal) takes over. But I would suggest not having excommunication, to ensure everyone can still be able to compete in the Church.
This EU3-cardinal system should probably just be streamlined by having different Church "sects" that can be influenced through bribes (that does not get replaced), whoever owns the most sects controls the Church, and that ties mean nobody controls the Church.