(3/X)
Crisis of the late 14th century, or The Pain of a Thousand Days
I think the Black Death would happen roughly as it did OTL, probably with slightly different dates (c.1350-60?) and transit methods. With Byzantium being a very coastal and heavily trade-involved state, it'll be messed up pretty badly by it but not ultimately destroyed. What it will do is tip the balance of the conflict in Anatolia in favor of the more lightly dispersed Turkmen, which will force the Byzantines to intervene directly to prop up and then conquer the weaker Seljuks. This will drive the Turkmen east, while leaving the Byzantines far weaker than they were before.
Byzantine infighting and entropy has reached meme-level for good reason. Over the entirety of its existence, despite the existence of several long-reigning dynasties (Makedonians, Komnenians, Megalokomnenians, and Palaiologians) the average reign of any dynasty was exactly 100 years (99.857, technically). This is due to institutional problems exacerbating the usual secular cycle, and as the institutions which caused this remain in place the cycle should still be working. As the Laskarioi have so far been portrayed as uber-competent (an error on my part), I find it reasonable for them to hold onto power about as long as the Makedonians or (Imperial) Komnenians did, generating a great amount of popular support and loyalty like that which sustained the pseudo-Macedonians. However, these good times cannot last, and by 1375 the cracks will be starting to show.
With an earlier !Timurid Empire, a large Turkish population with a grudge and an increasingly unsteady government in Byzantium, the storm is obvious. By the 1390s, the provinces are starting to drift out from under central government once more and several noble revolts and army mutinies have occured in Anatolia and Greece, with the existing Laskarids clinging to power through concessions and being too popular to remove. In, say, 1392, !Timur pacifies Syria and turns his gaze westward.
!Timur crashes through the eastern frontier, being greeted as a liberator by the Muslims of central Anatolia and capturing many of the strategic forts intended to maintain the eastern defenses. Sufficiently stunned, the basileus and the magnates put their differences aside and the basileus, let's call him Alexios VII for irony, gathers his army and marches eastward. !Timur crushes him and chases the remnants of the Byzantine army to the Hellespont and Bosphorus, where the Imperial Navy can stop him. A vast web of watch-fires, from the camp of 150,000 men, burns in view of Constantinople like the gates of Hell itself. He then lays waste to Anatolia, unleashing an orgy of violence and slaughtering Greeks and Armenians left and right, tearing every major city brick from brick sending waves of refugees fleeing to the islands or Crimea and unleashing a flood of Turkmen both new and old across Asia Minor. He then turns and sweeps northward, pillaging the Caucasus once more and sending the Orthodox peoples fleeing into the highest mountains before savaging the plains colonies, only being prevented from entering the Balkans by a desperate alliance of Mongols, Russians, Lithuanians and Poles. He passes through Anatolia in one last sweep, then turns and vanishes eastward, killed by one of his slaves in 1396. By 1400 his sons and generals have divided his empire amongst themselves, putting an end to the last of the Mongol conquerors.
The devastation is immense. Of all the Greek cities in Anatolia, only Attaleia, shielded by her high mountains, narrow passes and desperate sons, survives. Vast swathes of the interior are depopulated or completely empty, the European half of the empire is choked with refugees and the old thematic system has been rendered completely impotent. The Turks have come screeching back, carving out countless beyliks and tribal states on the corpse of a unified Anatolia. The empire is left in a succession crisis, as Alexios VIII died without an heir, and the entire state is paralyzed, almost in a fugue state.
It is a truly defining moment.
(P.S. If you think I'm exaggerating this, compare with OTL Timur's treatment of the Hindus in India. It may be slightly overdone, just because I'm tired of seeing TLs where the Timurids annihilate the rest of the ME while leaving the Byzantines intact 'because', but it is entirely within the bonds of plausibility.)