The challenge is to get the blue blooded Hapsburgs onto a Byzantine Imperial throne before 1600
(yes, that means you can have Byzantium last longer than OTL depending in your PoD)
Revised version of the above, hopefully without howlers.
Trivial. A junior prince of the House of Habsburg (Otto, born 1171 (PoD), younger brother of Count Rudolph II), joins the 4th Crusade. At the time the Habsburgs were based in now-Switzerland and Alsace, adjacent to Burgundy and Champagne, the source of most of the 4th Crusaders. (OTL, Rudolph's son, Count Albert IV joined the Barons' Crusade of 1239, and died in the Holy Land, so the Crusading impulse was present in the Habsburgs.)
Otto becomes a potentate in the Latin Empire. In 1205, the first Latin Emperor, Baldwin, is captured in battle by the Bulgars; his brother Henry is killed. (Butterfly effect - OTL Henry stayed out of the battle, became regent, and succeeded Baldwin when he died in captivity soon afterward.) Otto wangles the regency, and then arranges the succession of Baldwin's youngest brother Eustace as Emperor, with Otto becoming "the power behind the throne". In 1206, Otto arranges an alliance with the ruler of Epirus, Michael I Komnenos Doukas, and marries Michael's eldest daughter. (OTL she married Eustace.) Otto and Michael defeat the Bulgars, put down rebellious Frankish lords in Thessaly, and otherwise collaborate.
Supported by Epirus, the Latin Empire beats back Nicaean attempts to regain territory from the Latins; first in Asia, then in Europe. In 1215, Michael of Epirus is assassinated by domestic rivals (as OTL). Michael has no legitimate son (as in OTL); Otto claims the succession to Epirus as his son-in-law, defeating Michael's killers.
Eustace dies childless in 1220, and Otto becomes Latin Emperor. In In 1222, Nicaean Emperor Theodore I Laskaris renews the invasion of Latin territories. Otto defeats and captures Theodore, and tries to negotiates a long-term peace with Nicaea. Theodore refuses, until the Seljuks of Rum invade Nicaean territory while he is still a prisoner. He then agrees to Otto's terms. Otto's son Arnulf (born 1207) marries Theodore's youngest daughter Eudokia (born 1211); the marriage is actually in 1227. (OT, Theodore died in 1222; Eudokia married Duke Albert of Austria.) Otto releases Theodore, and sends troops to help expel the Seljuks. The alliance is victorious, and Nicaea gains Seljuk territory.
In 1228, Theodore dies, succeeded in Nicaea by his eldest daughter Irene's husband John III Doukas Vatatzes, who is supported by his in-law Emperor Otto. John has a son Theodore (born 1222), and a daughter Maria (born 1228; OTL, he had only Theodore). Arnulf succeeds Otto as Latin Emperor in 1236. When the Mongols invade Anatolia in 1243, and smash the Seljuks of Rum, Arnulf persuades John to join him in offering submission to Great Khan Güyük, and assists him in picking up Seljuk land. Nicaea expands to the Taurus Mountains, nearly doubling in size. Arnulf's son Conrad (born 1229) marries Maria Doukas in 1247; his daughter Constance (born 1232) is betrothed to Theodore Doukas.
The growing closeness between the Latin and Nicaean courts offends many Nicaeans, who regard the Latins as gross usurpers and yearn to avenge the sack of Constantinople by the 4th Crusade. Michael Palaeologus becomes the leader of the anti-Latin faction. When John dies in 1250, he is succeeded by his son Theodore II Lascaris, but Theodore alienates much of the court by favoring capable commoners (as in OTL). In 1254, Michael Palaeologus is detected conspiring with the Seljuks, and to avoid arrest and execution, stages a palace coup against Theodore, who suffers an epileptic fit at the wrong time (OTL he was an epileptic, though fairly functional). Michael poisons Theodore, gains control of Nicaea, and blinds Theodore's four-year-old son John (as he did OTL in 1261 when the boy was Emperor John IV).
Arnulf promptly marches on Nicaea, and seizes the city with help from Lascarid loyalists. He executes Michael, and enthrones John, with his sister Constance (John's mother) as regent. He then withdraws from Nicaea, despite the many Latins who want to take the opportunity to conquer Nicaea and eliminate the last significant Byzantine successor state. Arnulf's obvious solicitude to the Greeks arouses hostility among some of the Franks, especially the Venetians, whose representatives are practically co-rulers of the Empire. In 1255, Venetian mercenaries attempt to storm the palace and depose Arnulf - with, as it turns out, the support of the Pope, who is annoyed with Arnulf's tolerance (verging on embrace of Orthodoxy) in Constantinople. Arnulf thereupon stages a counter-coup with assistance from Nicaea, driving out the Venetians and any contumacious Roman clergy. Arnulf establishes an inverted "Uniate" Catholic Church in in the Latin Empire: it rejects allegiance to Rome, instead adhering to the Patriarch of Constantinople. This eliminates the influence of the Pope in Empire affairs. But it retains the Latin rite, which keeps enough of the Franks on-side (including some office-hungry clerics) to keep him in power.
Constance has political problems in Nicaea: she is a Frank, and the blinded John is regarded as therefore ineligible for the throne. When his father's epilepsy appears in him, discontent mounts further. John himself wants only to abdicate and enter a monastery.
When Otto dies in 1259, to be succeeded by his son Conrad, Constance proposes that John abdicate in Conrad's favor as well. Conrad is three-quarters Greek, independent of the hated Pope, and the enemy of the hated Venetians, and married to the senior descendant of John III Doukas.
In 1260, Conrad von Habsburg is crowned Emperor in Hagia Sophia in a hybrid rite, with nobles from Nicaea joining the Epirots and remaining Franks in swearing allegiance to The Emperor.
(OK, I
think I've got it clean now. I made more howlers, but this time I think I fixed them all.)
