By 1300, the Mongols have split off into the major khanates and the Mamluks have grown in strength in the Levant. The Mongols do not attack the Empire though, due to shrewd diplomacy conducted by the Emperor and Byzantine government (essentially paying off the Mongols in gold along with a princess or two. The Mongols also attack other countries like Bulgaria, which helps relieve the Empire as in OTL which benefits the Empire as it lessens tension at the northern border. Once the inept Alexios III Angelos dies, the Empire would, over the course of the century gradually expand in Anatolia by taking advantage of the Mongol invasions which devastate the region. Instead, the Empire continues puttering onwards without the Fourth Crusade. The premise is that the Fourth Crusaders never arrive in Constantinople, sparing the City and her inhabitants from the terrible and tragic sack that shattered the Empire.
This is an alternate history map that depicts the Eastern Roman Empire (Byzantium) and neighbouring states in 1300.