Okay, don't get me wrong. I like Stoicism. A version of the Roman Empire which embraced Stoicism as it's state doctrine would be interesting and something that I'd read a TL about. But that's not what this challenge is about.
The idea behind this is that some Roman philosopher pulls a Confucias and manages to codify the traditonal, disaparate rituals and beliefs of Roman society into a coherent philosophy/belief system. The challenge is to create a timeline where that happens and debate what it would be like.
That isn't really what Confucius himself did. Confucianism distinguished itself
against the other hundred schools of thought, and constantly fought against it's competitors. Neo-confucianism which came much later is when we see it start to embrace elements of the other religions, but still in an opositional context. Much of our texts on the various other philosophies of the hundred schools era come from Confucian texts rejecting them (often unfairly so). A good example would be Yangism, an egoist philosophy (which would make for an amazing chinese history) where we have one or two texts, the rest of our commentaries being statements along the lines that Yang Zhu would not lift a finger to save someone it it was a mild inconvenience to him (which is not true of the philosophy itself).
By the time Buddhism came along, the religion that Confucianism most adopted RE neoconfucianism, the term "naval gazers" was a slur which I beleive stuck to the modern day.
Now Stoicism, or at least aspects of it, could be part of this 'Roman Confucianism' since in all honesty Rome cribbed a while bunch of cultural stuff from Greece, especially Philosophy. But Stoicism itself wouldn't be the ethics system we're talking about,
My idea was more that confucianism wasn't that far away from developing into something like Confucianism. Looking at people like Seneca who, whilst one of the major figures of stoicism, felt it important to translate the Greek classics into Latin and wrote plays which celebrate Stoic virtue in a Roman context ("He who boasts of his ancestry praises the merits of another!").
What stood Confucianism apart was not just a worshipful attitude towards Chinese culture (which existed in Roman Stoicism), but a beleif in systematic learning through the beauracratic class (which did not exist, but I will get to this) and adoration of chinese classics. In the Confucian tradition, the veneration of the four books and five classics is significant, both in terms of popular philosophy and the ability to analyse that philosophy through texts. This attitude did exist in Stoicism but was never really codified as demonstrated by the Enchiridion in which Epicetus ends the book by referencing a wide variety of classics and philosophers.
In short, Stoicism to become the Roman Confucianism may just need official state support. If not directly to create a beauracratic class, perhaps some kind of mass education for Roman citizens which is Stoic based and perhaps the ability for the state to venerate individuals as Stoic sages alongside triumphs.