Lets say the Mongols are able to win at the Battle of Ain Jalut in 1260. Lets not make it a crushing victory, just a clearly Mongol one. How boned are the Mamelukes? Could the Mongols push into Egypt or were they already really at their limit? What affect would this have on the Middle East? Would the Mongols sweep Jerusalem and Tripoli into the sea also? How would that be received in Europe?
At Ain Jault only a single Mongolian corps (1 rumen plus Armenian and Georgian contingents) was engaged so even in the case of a complete success conquest of Egypt looks unlikely. Hulagu with a bulk of his army was away in Mongolia and after return got engaged in a war with Berke (ruler of the GH), which he lost. So he did not have resources adequate for the conquest. Anyway, it was argued that an absence of the good grazing areas did prevent the Mongols from a long term occupation of Syria.
Now, if the Mongols are victorious at Ain Jalut, it is anything but clear why would Kitbuqa, who was a Christian, do something nasty to Jerusalem. From the Christian point of view he could end up as a second Godfrey of Bouillon, leader of the Yellow Crusade.
as for Tripoli, it could end up as one more Mongolian vassal state.