Why would a lasting Byzantine empire establish colonies in the Americas? They have quite the opposite incentive, the same as the Italian states: they get rich off the trade to the East. Sure, they might get gifted a few small Caribbean islands like France did with Malta, but that's what they are.
I think if you want a Byzantine state without a wank, you would need the Mongols to focus entirely on the Middle East, not only destroying Arab civilization but stunting the new Turkish one.
Also, who said it would need to be based in Rome? With the Kievan Rus intact, the two merge into something like the Polish-Lithuanian Commonwealth, retaining the heritage of Rome and eventually moving to the Third Rome, Moscow.
I think if you want a Byzantine state without a wank, you would need the Mongols to focus entirely on the Middle East, not only destroying Arab civilization but stunting the new Turkish one.
Also, who said it would need to be based in Rome? With the Kievan Rus intact, the two merge into something like the Polish-Lithuanian Commonwealth, retaining the heritage of Rome and eventually moving to the Third Rome, Moscow.