Well, actually Alexius apparently was marching an army out to "help" the Crusaders take Antioch. However, deserters from the Crusaders' army told him that all was lost, and he turned his army around, never reaching Antioch. Thus, the Latin Crusaders argued, it was not they, but Alexius, who had broken the oath sworn in Constantinople. He was, as their liege lord, obligated to help them, and since he didn't arrive to help them at Antioch, he had broken the oath.
To achieve what you want Riain, have Alexius continue his march, and the Byzantine army break the Second Siege of Antioch, when the Crusaders are trapped inside the city. With Alexius having just saved their bacon the ruler of Antioch, at least, would have been a vassal of Constantinople. With the aid Alexius has provided at Antioch, the oaths of fealty that the Crusaders swore would still be in effect, since Alexius has kept his up his end of the contractual obligation. Thus the rest of the states the Crusaders establish would have the same obligations to Constantinople as Antioch. The history of the fall of the rest of the Holy Land may look much different. With Antioch having fallen, Alexius probably would want to go after Aleppo next, with its key strategic and economic position.
This may seem like it doesn't really matter, but it does. If the Crusader states right off the bat are vassals of the Empire, then you will end up having a closer relationship between the Empire and the Latin Outremer much sooner. In OTL it was a pretty acrimonious relationship, and the Latin Outremer only recognized its shared interests with the Byzantine Empire after the Second Crusade (1148). One of the things that is noted in both Zirgid and Nur al-Din's campaigns against the Latins is that they didn't want to get the Byzantines involved in the Latin-Muslim wars in the Holy Land. In this TL the Byzantines would be much more involved in the wars, because the constant border skirmishes would be between the Emperor's vassals and the Muslims, not between those crazy barbarian Latins and the Muslims (OTL). The Latins being in the Empire actually will probably end up securing Syria much sooner. The Empire would want both Aleppo and Damascus in the fold, and if those cities fall in the first half of the 12th century, then the Latins would be secure enough to use extensively in the campaigns against the Turks in the interior of Anatolia. The Rum Sultanate would be surrounded on three sides, with the Byzantines to the north and west, and the Latins to the south.
Furthermore, with the Latins as Imperial vassals, Riain, you'll be able to get what you want, an army of Latin knights fighting for the Empire.