Ok , this may be drifting a little OT , but let's suppose two things : 1 , no Mongol invasion . 2 : Everything goes right for the Crusaders . By 1200, a unified Crusader state stretching from the Tigris to Cyrenica has emerged , centered on Jerusalem . In game terms , in the hands of a seasoned player , this should be reasonably achievable. In real life , the Crusaders would have to be absurdly Lucky. Let's say that Mecca is also in the hands of Christians .
In such a circumstance , I can forsee that Iranian Muslims states would continually attempt to reclaim Islam holiest places . Are you telling me that a Crusader Empire , where everything is going right , will simply sit on the Tigris and not progress into Persia for 200 + years and simply soak up those raids ? Odds are , an Empire in such a position , by 1300 , if not earlier would be in an excellent position to expand into Persia , assuming no significant threats on the Western frontier. Not to mention that there's plenty of motivation to do so.
Of course , such a scenario would require so many things to be right that I don't think it's quite likely - but seriously , unless the CK2 AI becomes infinitely better than the rest of the Paradox titles , the mere presence of a human player not restraining himself would be akin to a string of lucky coincidences for the hypothetical Crusader state.
Finally , axing Iran will make simulating the Southern thrust of the Mongol Invasion quite difficult to balance . At least the Il Khantate horde tearing through Persia should be whittled down quite a bit before reaching Baghdad , let alone Jerusalem .