A Byzantine-Ayyubid alliance that brings results seems like a very good way to get an ITTL 4th crusade aimed at Constantinople, rather than the OTL one that hit it by accident.
More seriously, throwing the Latins to the wolves may bring short term gains on their eastern flank, but it invites a world of pain to come out of the West. That’s the Byzantine dilemma in this age, there are powers in all directions eager to take a piece out of them, and they aren’t quite strong enough to hold them all off. If they could gain consistent military ascendancy on one front, then they could hope to hold the other by diplomacy, but at this point the Byzantines can only manage to be a strong force on one front, not a dominant one.
This is a very true post.
OTL, the westerners were dismayed by the policies of the emperors that came after Manuel I Komnenos (died 1180). The Pope himself wrote a letter to one of the Angeloi emperors, urging him to "be more like your illustrious predecessor, Manuel Komnenos, of famous memory".
Despite his troubles at times, Manuel was highly-regarded by a significant part of western opinion. For example, one of the chroniclers of the Crusades, William of Tyre, called Manuel "a wise and discreet prince of great magnificence, worthy of praise in every respect", "a great-souled man of incomparable energy", whose "memory will ever be held in benediction."
Indeed, much of the anger directed at Byzantium in the years between 1180 and 1204 was precisely because the westerners knew that things used to be better in the time of Manuel I, and they were unhappy about the change of policy after his death.
The crux of the matter is that Byzantium needed to defeat one or the other of its enemies; either the Normans of Sicily must be destroyed, or the Seljuks of Rum must be conquered. Both continually posed a mortal threat to the empire, and as long as they both existed, the empire was constantly under the Sword of Damocles, a disaster waiting to happen.
Manuel, for all his faults, at least partially recognised this. He launched invasions of both states. The problem is he failed.
Realistically, the forces he deployed in the kingdom of Sicily were not going to be enough to achieve his aims, but he seems to have over-estimated the gains that large amounts of gold could achieve by way of bribery and mercenaries. The lack of a large enough fleet also led to defeat.
In his later years he seems to have taken action to rectify this, because by the later 1160s he could send an impressive fleet of over 200 ships to invade Egypt. But arguably these resources might better have been used to attack the kingdom of Sicily. It is notable, incidentally, that Manuel's fleet was sufficiently powerful to gain victory against Venice in 1171. A fleet of 150 Byzantine ships pursued the Venetian fleet of 120 ships across the Mediterranean. This would be one of the very last major successes of Byzantium at sea, and is an example of why Manuel's reign is such an interesting turning point in the history of the empire.
In the case of the Seljuks of Rum, Manuel launched an attempt to take their capital in 1176, but was defeated in battle at Myriokephalon. Although the actual fighting was technically a draw, Manuel's siege equipment was destroyed, and thus the campaign was no longer viable. Strategically therefore, it was a major humiliation.
Returning to the Byzantine-Saladin alliance, therefore, it seems that it would have been a populist policy which might have played well with domestic opinion at the time, particularly given popular resentment at the west. This is especially so, given the Norman invasion and sack of Thessaloniki in 1185 (after which the Normans were defeated in battle and driven out). It also offered the chance of swiping Antioch once and for all, thus settling the score and putting right one of the major Byzantine grievances against the west, which had been festering for a century.
Beyond that though the policy does set the empire on a collision course with the Third Crusade, which OTL they utterly failed to handle. Byzantium needs competent leadership to succeed. The trouble is by the time the Byzantine-Saladin alliance comes about, the empire is in a difficult situation. Bulgaria rebelled in 1185 and became independent, greatly reducing the empire's ability to project its power. Serbia (a vassal) also broke away, as did Hungary and the Crusader states (all vassals of the empire in Manuel's day).
Bulgaria rebelled because of Isaac's taxation demands. But perhaps heavier taxes were needed, because of the costs imposed by the Norman invasion, and the fact that Manuel had spent so much money on his various failed campaigns in Italy, Egypt and Anatolia, plus his successor Andronikos had been violently overthrown and the chaos had probably further depleted the treasury.
Ultimately therefore it seems an alliance with Saladin won't fix the core problems of the empire, which had built up over the last century. Byzantium needed to retake Anatolia to be secure; only then could it really afford an all-out war with the west. Short-term gains from the alliance probably played more to popular appeal than long-term careful strategic thinking. In the circumstances, it might have been wiser to continue Manuel's pro-west policy for a while, attempt to improve relations with Venice to keep the Normans in check, and focus on fighting the Turks.
For example, the Third Crusade took Konya in 1190. Had the Byzantines wholeheartedly allied with the west, instead of allying with Saladin, they might have been much better placed to occupy the city and hold it for the empire. That would have opened the possibility of major gains in Anatolia, completely changing the situation of the empire much for the better.
But it's easy to say that with the benefit of hindsight. At the time, the political situation and public opinion demanded a tough anti-western stance. So we end up back where we started; the empire is menaced on both sides by dangerous enemies, but lacks the ability to fight both at once. The massacre of the Latins in 1182 would have made it difficult to repair relations with Venice. Campaigns to defeat the Normans are impossible while the Turks are at war with the empire. But resources for the east are difficult to muster when Bulgaria is in rebellion in the west. The empire is surrounded on all sides by enemies and its resources are diminishing.
The only way out would appear to be a series of decisive military and diplomatic victories. But it would not be easy, whichever way one looks at it. That's why the reign of Manuel I Komnenos is all the more tragic. He had the wealth, the power, and the armies and fleets, as well as the time and diplomatic space to tackle his enemies. He just failed to do so effectively.
The empire would not get another chance.