Is there a way to prevent the rise of
Caesaropapism in the East?
Caesaropapism, as has been sort of halfway pointed out, did not "rise" in the East; it was the default from day one. From the point when Constantine summoned the bishops to the Council of Nicaea, it was very clear that the Empire was taking an interest in Christianity, and it was a long-term interest. There were bishops and churchmen who resented Imperial influence, and attempted to establish independent powerbases, throughout the Empire. The ones who succeeded were in the West, which is where you see Aurelius Ambrosius of Milan and his successors successfully challenging the power of the Emperor and laying the foundations for the independent Church hierarchy in the West, but this movement failed to gain traction in the East, so as East and West diverged, the West ended up with an independent Church, and the East with one in a generally separate, but definitely dependent to an extent, on the Empire. The term "caesaropapism" was developed by Western scholars who treated the Western Church's development to be the default, despite it being actually being the one that split away from the original norm that was retained in the East.
This wasn't guaranteed to be the case, but, really, the question is how to cause Caesaropapism to die out in the East as well as the West; not impossible, but rather a different task.
And even that's muddied, of course, because for all that the West lacked Caesaropapism for most of its history after the division of the Empire, the secular governing powers (meaning the non-Church governments, not religiously free/tolerant governments) such as the Franks, Lombards and Visigoths, and your later France, Holy Roman Empire and Aragon to replace them, certainly maintained a fairly large thumb on the scale in Church politics even with an independent Church hierarchy.