Personally, I tend to put in a period of division between the "full" Roman Empire, and the Byzantine Roman Empire, which comes down in the period 650-750. The Roman Empire before this is still recognisably the same beast as that existing in the time of Caesar and Augustus: afterward, it's been transformed.
So, I'd say the last chance to save the earlier Empire is to somehow butterfly the Great Plague of 542. Without it, and its devastating recurrences, the whole Empire is on far steadier ground even if the other problems of Justinian's reign still happen.
Regarding the later Byzantino-Roman Empire, the best point for a proper "saving" of the state is the POD I used for my own TL, to get a competent dynasty of Emperors on the throne sometime between 1040 and 1060. It's certainly possible to avoid the disaster of the Fourth Crusade, but I strongly suspect that a Komnenid sized Byzantine Empire would be extremelly tempting when the Mongols come along, and they'll finish it off.