Relations with Byzantines would be still tensed : rivalities on North Syria (as for Antioch, or Zengid principalities) were really an issue there. All depends how well Manuel does in Anatolia, and how well Baudouin does in Syria.
Honestly, I don't see the war between Nur ad-Din and Shiruk from one hand, and Franks from the other being easily butterflied. Maybe no initial expedition, that may have been due to Amaury needed some strong military fact to strengthen his position on the throne.
But sooner or later the tribute was going to be disnounced by Egyptians, and Baudouin would need it to finance his defence and Fatimid Caliphate going trough a large decline was simply too tempting to not act.
Maybe Baudouin would be able to make his vassals waiting for Byzantine help ITTL, something that Amaury wasn't able to do, and make a more successful campaign in Egypt, turning it into a semi-protectorate of Byzantines and Franks.
Eventually, it would be hard to make both sides working really together (in spite of the more or less acknowledged byzantine suzerainty and warmer relationship), but you could butterfly away Saladin this way, and have (depending on the Anatolian situation) a more unified front against Syrian Zengids.