If Roger Crowley is to be believed, your PoD could be the successful defense of Constantinople (as in, a few more days) by Constantine. The viziers were on the cusp of civil war, Mehmed is meant to have been at risk of being overthrown for the sheer number of resources thrown at this conquest.
Cue a civil war of the Ottomans, which crucially leaves Constantinople in Christian hands (and depending on the chaos in detail) in a position to take some small advantage of the civil war to strengthen themselves, perhaps retaking Athens and some other territories. I'm personally a fan of "It tears the Ottomans apart", so lets work with minor PoDs between the Viziers and the Ottomans subjects to ensure that happens, so free Bulgaria, multiple Turkish Beyliks, etc.
This opens the door for potentially a marriage, or an alliance of this Neo-Roman state and (say) Russia. Now just to ensure you have them both in the same Empire, and also to grant Moscow the title of 3rd Rome, have a Russo-Roman dynasty replace/become the Palialogoi, and declare Moscow the Third Rome, effectively naming the Grand Duke of Moscovy the Co-Emperor of Rhomanion in the North. (i.e. just a rebranding there, effectively Russia has this Roman state as a satellite state that it supports to ensure access to the Med, and to secure its southern flank).
At which point you've got a stronger Russia, no Ottomans, and someone able to effectively bankroll a new Roman Period. Russia can threaten Poland in the East, and the Romans can from the South, completely different armies, two different campaigns, splitting the Polish, or after that, the HRE - allowing the Romans to march into Italy, whilst Russia would be able to ensure two-front wars.
Eventually you'll have someone try to rule all of them from one place - and the likely place is Constantinople, or to please the Russians, Sevastopol. The Black Sea really is a potential beating heart for this union, as assuming the Caucauses are taken at some point, they'd jointly rule it, and it'd be a major population hub, fed by the Black Soil of Ukraine, it would also be one of the most secure regions of the union as well, as it is deep in their territory, or behind mountains. So both of these cities makes sense.
As an aside, this union is probably capable of uniting the entire of Europe by force given time - which solves their mutual security problems. The geopolitical threats in this scenario are the people in the Eurasian Plain, excluding the Russians, but with the ability to fight (albiet disjointedly) on all their fronts at once. A European conquest (and Med) basically leaves no land border to be concerned about, instead it relies on securing seaways - i.e. Britain, the Baltic and its rivers. This could be Europes "China".
TL;DR - Save the Romans long enough for Russia to become Co-Emperor/Co-Empire. If stablised, the future for a Roman-Russian Union would likely point right at your challenge. The desire to unite the Three Romes. (Hell, you could actually achieve that, and have the WRE, ERE, and NRE, even if the NRE is called the Russian Empire).
(I'll admit there is a LOT of minor PoDs needed to make this work, especially just to have the Romans survive. I'd personally go with Mehmed and his heir dying rapidly after 1453, perhaps in a poorly-supported coup - but I reckon it ticks all the boxes).