The 1000 start date is the easiest to blob as the Byzantines. The 867 is the second easiest one, but it can be arguably easier than the 1000 one if the Abbasids explode, so that you can get South Italy, the Balkans, the Caucasus, Syria, and Palestine all during the course of one ruler. But 867 leads to way too ahistorical outcome in a long campaign since there are a lot of nomads and tribes, and they are always unpredictable.
The 1000 start date also provides you with ample opportunities to create realistic enough alternate history, since Christianity is still somewhat united. But some people don't like the schism mechanic since the outcome can be erratic. I don't encounter such problem personally. I find Carolingian restoration to be realistic enough here, in 1066 and beyond they are on the brink of extinction and you have to rely on dice roll a few times, I'm sure. If you like to be a masochist, go ahead though.
In 1000 you will often see the Iberian peninsula fully under the control of the Umayyads within the span of a few rulers. If you want to see interesting holy war shenanigans, 1066 start date is better. Also, 1066 is far better for Sunni Islam in the Middle East since the Seljuks are already in power.