I did the Bulgarian game last week, and I didn't have too many issues - I sent all my cavalry in a mad rush to Constantinople, where Justinian moved one of his two archers out of the city (???) as I approached. Needless to say, Byzantium fell extremely quickly to me. I was a little shaky as I moved to the west, but I never conquered or settled until I was at +2 or 3 and could afford an unlucky stability swing. I was more concerned with getting the faith points, but in the end, I wound up with enough for 110 (I reached 100 3 turns before the time limit and popped another great prophet on the turn after that).
I agree that the time limit was tight, but it was certainly doable if you reserve your settlers and conquests until your stability is ready for it.
By the way - I know the Cordovans respawn as the north African moors after they've been killed off. Will the Bulgarians respawn as the Second Bulgarian Empire if conditions are right? I'm 2/3 through a highly successful Byzantine game right now (all local enemies have been killed - Arabs, Bulgarians, Ottomans) as well as the distant enemies (Genovese, Venetians), but I didn't see a respawn of the Bulgarians (likely because I had a welcoming party strong enough to take Preslav on the turn it was founded - they had time to build no other cities).