Why should the crusaders give up Jerusalem?
How should the Pope govern Jerusalem? It would be more reasonable to give the city to a local church leader.
You need a really religious leader.
I'm currently not much informed about the crusades and their leaders, therefore can't recommend a detailed PoD, but I see two different possibilities, either change the conquest during the first crusade, or change the structure of the crusader states after the first crusade but before later defeats of the crusaders.
You could either change who of the leaders of the first crusade controls which town, or bring the first crusade after Antioch but before Jerusalem into failure. After that, the crusaders need to control Antioch for some time as their political centre, but conquer Jerusalem which is now not the capital.
Jerusalem after the first crusade is depopulated, therefore it would be reasonable to make the important harbour town of Acre sooner the capital of the Kingdom of Jerusalem, but a change of capital would perhaps mean a loss of prestige.
A solution would be to give the town the Patriarch of Jerusalem but using it still as a formal capital but using Acre as a political centre. After that, a personal union between Jerusalem and Antioch could lead to Antioch as a political centre.
If you want to make Antioch the capital of all the crusader states, you need to unite the crusader states and prevent all disagreements between them.
Also Antioch has the disadvantage, that a restrengthened Eastern Roman (Byzantine) Empire could try to recapture it (as in OTL). If Antioch is under Byzantine control, this could lead to chaos and balkanization of the crusader states or in the Crusader States becoming vassals to the Byzantine Empire.
In the long term, even an unified crusader kingdom has not much chances to survive in the long term.