Do not let their large territory and large armies fool you - it doesn't take much to plunge the Caliphate into a massive civil war, which will completely divert resources away from an external threat. Now obviously the Republic had been gone for centuries, but the Senate based in Constantinople was still a powerful institution.
When you create the Holy Sepulchre and arrange it in such a way that the title pops back to you on the death of the Grand Master you get ALL THE MONEY - this happened a few times for me but only with male heirs, never with female heirs and not all the time with male heirs either. Let's just run down a checklist for what are a few basic tenets for what a centralizing empire would strive to do. Just buy favors from a few people and you can force them to vote for your candidate. The Byzantine Empire is one of the few empires that is already created at both the 8 start dates. Also, invest in your retinue when you can - this increases your power relative to your vassals, and is a big deal.