Make it possible for the Latin empire to survive in Constantinople along with the duchy of Athens and kingdom of Thessaloniki. They must survive till the mid 16th century.
Unless the Muslims go the way of Grenada, i.e. splintering and spend all their energy fighting each other and the various Greek powers grasp the idiot ball firmly then the Latin's are doomed, tiny minorities can manage to take over an existing state, look at Anglo-Saxon England but not when there are multiple external enemies, then the inability to rally the serfs to the defense of the realm is deadly.
Anglo-Saxon England is not the best example, given they drove most of the minority to Wales. A better example would be the Norman conquest of England...But I digress.
It is not wrong to say the Latin's are doomed, what with the Nicaeans, the Bulgarians, and even the Orthodox Greek population. The Muslims (which for this situation, the immediate threat is the Seljuk Turks) while is a credible threat, is not an immediate one. The first thing any Latin emperor would have to do is to placate his orthodox subjects, which is a thing that might be really, really hard to do.
I meant the Norman takeover of Anglo-Saxon England.
Actually what if one of the Latin Emperors dropped the Latin thing i.e. decided that Constantinople was worth a Greek Orthodox mass? Assuming he managed to keep his Latin vassals in line and his head on his shoulders could a Orthodox Frankish dynasty survive and maybe even reunite the ERE.
Finally, would it be possible to increase the available manpower of the Latin Empire by a migration of most of the Normans from Southern Italy to the Latin Empire if it looked like they had no future under the Hohenstaufens?
The beginning of the end was the capture of Peter of Courtney en route to succeed Henry of Flanders. The Latin Empire has a better shot if he reaches Constantinople and has a reign like his predecessor.
Longer term, the fragmented Latin principalities and the equally fragmented minor Greek empires cannot continue in their 1204 form for too long. Somehow Nicea and Constantinople must combine forces and eliminate their most dangerous regional adversaries (eventually nipping the Ottoman Empire in the bud).
Finally, would it be possible to increase the available manpower of the Latin Empire by a migration of most of the Normans from Southern Italy to the Latin Empire if it looked like they had no future under the Hohenstaufens?