I don't think that's a given.
Yes I know the Catholic church was rotting from the inside at this point, but it wasn't about to splinter and fall apart at with the slightest blow. The Papacy had moved out of Rome before, though its reclamation would be a major goal.
With the fall of Italy, the Roman Catholic Church might reform itself come the Reformation in this timeline.
Here are some reasons why:
-the Pope isn't stupid, and he knows what will happen if he stays in Avignon. The Popes spent the whole Italian wars fighting off Austrian and French attempts at hedgemony, so why would the Pope move somewhere with a fast reach of the French crown.
-It might actualy safer for them to relocate to one of the electoral Bishoprics, allowing them to keep the French at arms distance and keep an eye on the Holy Roman Emperor.
-Of course Roman finances are going to be ruined with the loss of the Papal states.
-this would probaby up the amount of indulgences being sold, but the increased resentment of this will be slightly mitigated to the Germans by the fact that the Papacy is now IN germany
-of course this is still all going to boil over at somepoint, and someone is going to get uppity
-if the Papacy is closer to the reform area, and Germany is really the only place it could happen due to its decentralisation, then a potential luther getting away with heresy becomes slimmer. People sometimes forget that Luther was REALLY lucky. He was called before Emperor Maximilian I, but before Maximilian could get his hands on Luther the emperor kicked the bucket. Then everyone stopped caring about Luther with the contest between Francis I of France and Charles V both going for the throne. This then led to Francis I just throwing money at the protestants to piss Charles off and Charles could never really get around to dealing with the Reformation effectively since he had to be everywhere and ended up no where.
-with the papacy closer at hand, they'll be able to root out any famous heretics really fast. This doesn't however stop the reform bubble from bursting and the Pope will probably be forced to call a council along the lines of trent
-now in OTL the Pope's were fearfull of convening councils because historically the councils had moved to reduce papal power. In OTL, the Pope was able to prevent block voting amongst the bishops and to use his power over the Italian bishops to ensure whatever he wanted he got
-now ITTL, the Pope is outside of his historicall power base and all the crowned heads of Europe are interested in taking the Pope's power down a notch. TTL's council might not go the way the Pope wants to and make a few concessions to OTL Protestant ideas that wouldn't occur till Vatican II, namely mass in the vernacular, increased laity participation in the Eucharist, and MAYBE, just MAYBE, marriage for the lower clergy.
-the European monarchies would probably also be able to get some concessions regarding increased influence over their churches.
Just thought it would be nice to suggest something different from " the church is going to shatter the moment rome falls."