Robert M. answered 04/18/24
Tutor, Edcucator, Historian, and Archaeologist
Terrific Question!
Let's start by discussing at what Christianity looked like at the time of Clovis' conversion. Now, we don't see an official and permanent division of Eastern and Western churches until the mid-11th century CE, but there was a powerful distinction between Christians who followed the imperial-sanctioned Nicene Creed and those who did not. In short, Clovis was not making a simple "yes or no" decision when converting, but was also choosing between competing interpretations of Christian belief!
So why would Clovis, given that he could have chosen a different sect of Christianity, adopt the one that tied him most closely to the pope and to Rome? We know that the spiritual relationship with Rome was maintained and expanded by later Merovingian kings and even more so for the following Carolingian dynasty.
One of the most significant benefits, particularly at a where the literacy rate was plummeting, was that good relations with the pope provided a steady supply of well-educated priests to serve as administrators, advisors, bishops, and chroniclers. As a ruler of a brand new kingdom with no prior tradition of literature or record keeping, Clovis would have seen the arrival of literate men eager to support his rule as (forgive the pun) heaven-sent!
You made an excellent observation of how the Frankish kings used Christianity to justify their wars of expansion, these are some of our best examples of how both king and church benefited from their partnership. The motivation for this, sadly, was that the use of religion to justify invasions made the conquest and subsequent establishment of Frankish rule easier. The Franks could depend on the church's moral authority to present their conquests and governance as legitimate and moral, and the appointment of bishops to convert and govern the conquered population expanded the authority of the church in Rome.
There was no way for Clovis to have predicted that his conversion would begin a process that would lead to a pope declaring Charlemagne as "Emperors of the Romans" centuries later, but he knew a golden opportunity when he saw one. Had the Western Roman Empire survived, or it's Eastern counterpart maintained it's hold on Rome and its influence over the pope, the Franks may never have had the opportunity to become protectors of the Roman Church and reaped the benefits. The job would have already been taken!
Long story short, the Franks would have benefited far less from allying with a powerful Rome. In fact, a less influential Papacy allowed the Franks to take on and redefine the role once held by the Western Roman Emperor!