(2/X)
Things change sharply in the mid-13th century as the Mongols roflstomp the Rumites, Bulgarians and Georgians and crush Byzantium's main rivals. I've heard speculation that the post-Byzantines survived the Mongols because they were too small/poor/isolated to bother with, but this seems suspect so I'll ignore it. With this sudden blow, Byzantium's foreign policies change dramatically and (assuming centralization and reform have progressed to the point where it is capable of foreign offensives) a return to the Komnenian Golden Era becomes truly possible.
The 2nd Bulgarian Empire was a deeply unstable state, with constant noble conflicts and extreme social tension (2nd Bulgarian Empire was the site of the only successful peasant revolt in European history), so once it is devestated by the Mongols it should be relatively easy to absorb. The technique is what matters, as a clumsy approach could easily unite the Bulgarians against Constantinople and draw out the conflict or, worse, drive it into Mongol tributage as a protection. The best approach, I think, would be to portray the basileus as an Orthodox ruler come to free the lower classes from noble tyranny and chaos, restricting the expansion of the pronoiai to secure a somewhat peaceable region and (ideally) a loyal base of men and money. No matter how it is done, this will take a time, likely more than a decade to truly pacify the region and half a century to truly integrate it, but if done well it will strengthen the Empire immensely. The Vlachs should be left as an independent state to shield Bulgaria from the Mongols, and everything should be done with great care to not provoke the khans.
Serbia was undergoing a period of domestic turmoil and civil wars (basically the entire Nemanjic Period), so the factions should be played against each other until Serbia is weak enough to be dominated. Care should be taken not to provoke the Hungarians or Venetians into intervention, however, as that could turn an otherwise backwater buffer into a cold war.
In the east, Georgia should be in flaming ruins after the Mongol invasions. The Byzantines should begin to establish political and military ties with the Georgian remnant states to increase their region and secure the Black Sea, with the eventual goal of establishing vassal states along the buffer and securing the warlike tribes of the region in service to the basileus. Existing Georgian missionary programs in the North Caucasus should recieve full aid as an important way of increasing support in the region, securing the support of the warlike Circassians, Vainakhs and Avars, future farmland and a buffer with the Mongols. It also helps protray the Byzantines as the center of the Orthodox world and increase cultural hegemony.
Finally, there are the Turks. This should be easy, actually. After the Mongol invasion, many Turkmen groups followed them from Central Asia and began settling in Anatolia. They, being truly Turkic fanatics, had little in common with the more Persianate and religious complex Seljuk Turks, and so in OTL and ITTL they set about a series of wars over land and culture. The Byzantines should play them off each other, slowly encroaching on the edge of the plateau with fortified settlements and networks of defense. This will take decades, if not centuries, but with a continuous program and rational(ish) actions it can be done very easily. Special care should be taken to play the Alawites/Armenians/'softer' Sunnis against the hardline Hanbali Sunnis supported by many of the Turkmen to intensify the conflict and ideally keep out outsiders. Eventually the two factions will have bled themselves, and as the Seljuks are far more familiar & reasonable than the Turkmen, the Byzantines should move in with overwhelming force to crush the Turkmen, kill all the men, sell the women and children into slavery and move various Greek, Albanian, Armenian and other Christian herder groups into their pastureland to make sure that they can't come back. The Seljuk nobility were very hellenized OTL, so with a stronger Byzantium and a longer and more galvanizing conflict with the Turkmen they should be even more so. They should be pulled into the existing Byzantine aristocracy as well as possible while all of Anatolia is conquered and incorporated into the Empire--a light hand should be used in much of it to keep the Sunnis and Armenians from revolting, but if this has been done right they'll be happy to just have peace after decades of war.
If everything goes well, by 1400 Byzantium will have expanded to include all or almost all of the Balkans and Anatolia, similar to their borders at the time of Basil II (minus Italy & Outer Armenia) and will have vassals stretching across Serbia and the Western Caucasus. Enter Timur.
Especially since in his later years he was a fiercely fanatical Muslim expansionist and went to great efforts to crush non-Muslims states - and those Muslim led ones who were more tolerant. Byzantium here, assuming of course a Timur figure still occurs, would probably be his target earlier than the Ottomans were OTL.