The problem with your theory (and it's really more wishful thinking than a theory) is that there is no evidence of a schism before 1054. Neither is there any evidence that a new church sprang up around the time of Constantine. There was one church before Constantine. There was no schism during the reign of Constantine. There was one church after Constantine. It was the same church before and after, except it was quite a bit larger after Constantine.
If there had been one church before Constantine and two churches after Constantine you might have a credible theory, but there were not.
Every church historian with any credibility disagrees with your position. Bruce Shelley (Baptist) and J.N.D. Kelly (Anglican) to name a couple. I've heard Leon McBeth (Baptist) takes the same position although I haven't read any of his books and can't say for sure.
First, this is not a theory. This is fact based both on the Bible, and thus the theology of ecclesiology, and then on historical records after that.
Theology first.
Ekklesia is the Greek word for church. It means assembly. In no case is it translated "The Church," nor can it have that meaning. It always means an assembly or congregation. It is correctly used in its secular sense in Acts 19.
Act 19:39 But if ye enquire any thing concerning other matters, it shall be determined in
a lawful assembly.
Act 19:41 And when he had thus spoken, he dismissed
the assembly.
--There were many of the city of Ephesus that assembled in the theater at Ephesus for a definite purpose. They were gathered together for one purpose. They were all there in one place.
Notice how it would have been impossible for this to be a universal assembly, or an invisible assembly, or a world-wide assembly, or a monstrous organization such as the RCC. It is always a local assembly such as it was here.
The word ekklesia always means local church, congregation, assembly.
Theologically speaking the RCC is not a church, it is a business.
There was no Church in the first century, only "churches."
Paul went on 3 different missionary and started over 100 local churches all autonomous one from another.
He wrote 13 books, all written to either churches or pastors of churches.
In Revelation Jesus penned seven letters addressed to the pastors of seven churches.
God is using the local church in this day and age. It is his institution that He has blessed, not the monstrosity business organization that calls itself the RCC but has no right to call itself a "church" for it isn't one--not theologically nor historically.
Others on this board are correct. It did not come into existence until the fourth century until Constantine made his false profession, and married the state with Christianity so-called. He used it for its own purpose much like Calvin did with Christianity of his day. He also married Christianity to the state and began his own state-church. Both failed.
In Constantine's day, he introduced idol's into the church buildings. He paganized Christianity. Paganism became Christianized and Christianity became paganized and the two worked hand in hand.
To Constantine's chagrin he could not bring the unity to the empire that he so wanted. There were always those that opposed it.
There were the Waldenses, the Montanists in Africa, the Cathari, and many others who always protested against the idolatry and corrupt practices of this so-called Catholic and pagan church.
It never began with Peter, and every Christian knew that. Its claims were false. They still are false. It is a pariah and blight upon the true name of Christ and His followers. The RCC does not even know what the "way of salvation" is. Their path leads to hell and not to heaven. It is opposed to Christ, and what Christ said in His Word.
Isa 8:20 To the law and to the testimony: if they speak not according to this word, it is because there is no light in them.