I think if Manuel I holds on for another couple years until his son is old enough to rule on his own you have a very different timeline starting around 1180 AD. A continuous transition of power from Manuel to an adult Alexios II would most likely have prevented the most catastrophic event in late Byzantine history, the sacking of Constantinople during the fourth crusade. A Byzantine empire that isn't weakened by the fourth crusade is able to weather the storm of the Hungarians, Normans, Turks until the Mongols come ( in Europe 1240 and then again in Anatolia 1400 ) and take advantage of the ensuring power vacuum to take out the sultanate of rum and the fatimid dynasty.
Here's how it might look: During Alexios II reign Byzantium would be able to continue offering help to the Crusader states helping them to limp along and continuing Byzantine influence within those areas ( maybe even reassert formal Imperial authority ). He would probably continue the pro western policies of his father, continuing to aide the Crusader states and would probably be viewed as a protector and friend of Christians. With a legitimate emperor who was viewed as a protector of the Christian faith, its unlikely the fourth crusade gets diverted ( especially since the casus belli of the diversion of the fourth crusade was to put Isacc II back on the throne, who in this timeline would have remained a loyal general ). But in general the empire would still be hard pressed on all sides and economically hard pressed from all of Manuel's ( and Alexios' II ) wars at this point; but still held together under a legitimate emperor, Alexios II. The arrival of Genghis Khan in 1240 would grant the Byzantine Empire a reprieve on its western front; allowing them to take back some of the territories they would have lost to Hungarians / Normans and recover economically. The continued presence of crusader states supported by a unified Byzantine Empire however helps slow the tide of the Sultanate of Rum during this time. Most importantly the prestige of the Eastern Roman Emperor ( assuming Alexios II doesn't die young and has a son ) would still be intact and the city of Constantinople would economically be independent of Venice and the other Italian states thus ensuring revenue for the mercenaries that the Byzantine army relied so heavily on. Between 1280 and 1380 would probably be rough for the Empire internally; the constant warfare and the end of the Komnenoi dynasty ( cause lets face it anything over 200 years for a Byzantine dynasty is way too long ), and a likely explosion of religiously and culturally fueled anti-western feeling would probably lead to some civil wars and unrest. So that by around 1400 the Byzantine Empire would territorially be diminished on its eastern front and economically exhausted. It would still have a decent amount of its European possession due to the Kingdom of Sicily ( Normans ) being absorbed into the HRE around 1200 ( thus no more invading Normans ) and the second Bulgarian Empire never forming ( no Andronikos or Isaac II or political instability during that time that would lead to a revolt ) so that a re-emergent Hungary would be the only direct European threat. In the east the Crusader states would be either destroyed or essentially vassals of the Sultanate/Fatimids, allowing the Sultanate to focus all its energy on capturing Constantinople, a war which the empire wouldn't fair well in. Their salvation however would come in 1400 when Timur invades Anatolia and practically wipes out the Sultanate of Rum. The main difference in this timeline opposed to the real one is that there is an economically independent core of Byzantine holdings such that the Byzantine empire is then able to take out the severely weakened Sultanate and take advantage of the turmoil to reassert its control of its former territories up to Antioch. The Ottoman Empire is never born, and the Byzantine Empire restores Imperial control in most of modern day Turkey and Antioch. They could even continue south to Egypt and topple the Fatimid dynasty which itself was experiencing its own difficulties around 1400-1415 ( a boy king, severe famines, plague and revolts ).
So by 1415 you could have a Byzantine Empire controlling most of the original lands it had during ancient Roman times ( albeit some more tenuously than others ). While it probably won't hold all of these territories forever, it would have endured long enough to ensure it survives as a nation into modern times.
First, thank you for breathing new life back into this topic. It's great to see new ideas being suggested
Alexios II is a somewhat problematic candidate for an emperor. The Byzantine historian Choniates wrote that Alexios was a vicious and insouciant youth. When papers would be presented to him, proposing that Mount Olympus be moved from one place to another, or that the heavens and the sky should swap place, he would sign them.
We cannot be sure that he would have been a good emperor, and the odds are against it (given that Alexios I, John II and Manuel I had already provided three good emperors consecutively, ruling for 99 years between them).
You're right that avoiding Andronikos I's usurpation of the throne in 1182 would probably mean the Fourth Crusade doesn't happen. That does have potentially major consequences for the region. But I think the Byzantines conquering the whole of Anatolia, Syria, Palestine and Egypt is unrealistic, and would require the intervention of Cheroptera Caelis Alienae.
Anti-western sentiment was rising and the behaviour of Venetians, Normans, Crusaders and others was the cause. By the late 12th century, it was clear to most people that these people were Enemies, not friends, and it was time for a fundamental shift in policy. There is no coincidence that the Angeloi sought to ally with Saladin against the Crusaders. They failed disastrously to gain any benefit from this policy, which brought them only shame and infamy. But that was largely a result of their own weakness and incompetence.
The point is, allying with Saladin against the Crusaders was actually a sensible and rational policy, and one rather more in line with the empire's interests, than the pro-western policy of the previous emperors. The Crusaders were treacherous, unreliable and frequently attacked the empire, as with Reynald de Chatillon's brutal ransacking of Cyprus in 1155. They betrayed the emperor John II Komnenos during his expedition to Syria in the 1130s, sitting around playing dice instead of helping at the siege of Shaizar, and refusing to hand over Antioch, using every underhanded trick in their ability to prevent the emperor from exercising his rightful authority over the city.
Crusaders passing through the Anatolian lands committed acts of theft, rape and destruction on the local Christians, generally making themselves thoroughly unpopular. It was time for a reaction. Powers such as Saladin's Ayyubid empire were civilised by the standards of the age, and sophisticated Arabic, Kurdish and Persian peoples of the east had far more in common with the Byzantines than the barbaric westerners.
Theorising about what happens after 1400 is unrealistic imo as that's 220 years after our point of divergence - much too far to say anything particularly meaningful with any certainty. I think what is plausible is that, if the Byzantine Empire can hold together internally and somehow continue with at least reasonably competent leadership, it can largely maintain its existing territories in Europe.
The real attraction of an alliance with Saladin is that it may offer the prospect of gaining control over Antioch, which had been a central aim of Komnenian foreign policy for over a century. There may even be a chance to seize a port or two along the Levantine coast, if the Byzantines act quickly. We know that Manuel I Komnenos had built up a large and powerful fleet, as he sent it against Egypt in the 1160s and again in the 1170s with over 200 ships, and also defeated the Venetians in a naval battle.
Looking at Anatolia, the question becomes harder to answer. There's no obvious benefit to a Saladin alliance there, other than perhaps leaving the Byzantine flanks relatively secure from that direction. It may be that by the 1170s, a strand of opinion had built up that it was time to end the dubious "vassalship" of the Seljuks at Konya and re-assert direct imperial rule here too. Manuel's policy of appeasement/overlordship most signally failed here, which is what led to a change of policy in 1176 and the invasion towards Konya. However, as we all know, his over-confidence/lack of caution led to defeat at Myriokephalon. It's an open question whether the empire could have launched another attempt in the following years.
The fate of the empire, as always, depended ultimately on three things:
1. Internal stability
2. Foreign policy (diplomacy)
3. War
Best case scenario, the Byzantines
might be able to get back Anatolia, although it will likely take decades of campaigning and a good emperor/series of them.
The internal stability nut is probably the hardest to crack though. The Komnenoi had turned the empire into a family business, and the imperial clan had grown too large and unwieldy to rule effectively. Corruption and decadence had set in. Perhaps another frugal, soldier-emperor like John II could have changed course, kept his family at arm's length, and put in the necessary years of campaigning. But it's a big "If".