Well, the last "Byzantine dynasty" ended with the Ottoman takeover in 1453...though it was never quite as dominant in the region after the sack of Constantinople during the Fourth Crusade. In particular, since Constantinople was just beyond the edges of the Mongol empire (but was very involved with international trade), there was a lot of exchange going on, especially with the Turks (meaning a Byzantine persona would be able to mix fairly easily with just about anyone within spitting distance from the 13th century without even worrying about period-compliance).