As there are multiple, sometimes merged and demerged threads on this subject, I am a little confused, and ask your forgiveness if you have already read my opinion.
The Lateran Council's importance can be exaggerated. Note that the second-to-last Ecumenical Council was held in 768, in, of course, Nicaea, not in Milan or Paris. In 769, the Church was in flux; the traditional major Patriarchates were in disarray because the Muslim conquest of Antioch, Jerusalem and Alexandria a century or so earlier was obviously not going to be reversed easily. But the schism on icons was hardly the only one; practices varied greatly, especially in the west. Not even Frankish bishops followed the Pope. The eastern shield and dominance had suffered a massive blow, but at the same time, the west had seen the rise of Muslim Andalus and the probability of another Augustine from Hippo was very low! There was certainly no uniformity even in the West; the Pope had plenty of problems with the uppity "patriarchs" of Aquliea and Grado, as well as errant bishops the length and breadth of his patriarchate. If anyone had asked a western bishop whether the Pope was infallible, the only vicar of Christ or whatever, he would probably have been intrigued by the notion. There was no investiture controversy because no-one would have taken the Papal claims seriously. The situation of the Papacy in 769 was precarious, and would remain so for a long while.
On the other hand, the recent downfall of the Exarchate of Ravenna certainly meant that the Popes had to fend for themselves, and naturally, they began to cut their ties to the Emperor in the East. However, I cannot see how an argument can be made for a permanent East-West schism in 769. Had the Pope presumed to send a legate to Constantinople addressing the Emperor as anything else than Emperor of the Romans, he would have been castrated. The Franks were hardly stable. The Lombards were a menace. The 800 coronation had not happened and there was no dubious Empress in the East. The Donation of Constantine had probably not been forged.
One could argue that even at the 867 start, there was no formal division into Orthodox and schismatic Roman; however, because of the rise of Charlemagne and his coronation, one could argue that the political position had irretrievably changed and that the gradual religious differences would inevitably follow.
However, at the 769 start, the political situation in the west was also fluid. One doesn't know whether Carloman would conveniently die in 771. If he hadn't, his elder brother's alliance with the Lombards might have pushed the Pope to support Carloman or ask for help from the ERE. In any case, there would have been no unified Frankish realm. So you would have had the situation in the 867 start, except that instead of the heirs of Charlemagne disputing his inheritance, it would have been the heirs of Pepin. And there would have been no WRE or HRE or whatever. Without the protection of a powerful realm in the West, the Popes would have been weaker, and certainly unable to act as freely in religion. They might have had to bend the knee to the Lombards, and then things would have become interesting.
So, no, for my part I do not believe that a Catholic / Orthodox division is sensible in 769 (Paradox's thinking about the Lateran Council notwithstanding - there were Patriarchal councils aplenty in the early Middle Ages, they often were at odds, but in the end, things were resolved at Ecumenical Councils, not because the Patriarch of Antioch or Constantinople or Rome decided things in his own little get-together.)
And there are a number of sensible and interesting suggestions in other threads as to how the Schism could develop.
I also think it odd that Paradox, having moved years ago from fixed determinism to conditional events, would not apply that here.
EDIT: More verbiage follows:
It is dangerous to look at events at a remove of 13 centuries and then decide that those events were inevtiable or could not be reversed. Consider the following: a games developer in 3100AD, sitting in his office (however that would be constituted) in Ny Stockholm on a planet orbiting a distant star, might create a game to model the rise of humanity, Terra Universalis. He might decide to start the game in 1867, because that's when Das Kapital was written. If he decides at that point to make 90%+ of the countries on earth republics or democracies (of one form or another, not necessarily in substance) because that is what happened 90 years later, and "it was inevitable", he would be getting the wrong end of the stick, no? And it certainly would have surprised the hell out of most people in 1867. Not only is Das Kapital not directly related to the death of monarchy, it is certainly not a pivotal reason for the development of representative government. The Lateran Council of 769 could well have become a footnote, as so many other western Councils.
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