One thing to keep in mind, Rome/Byzantium wasn't as big on dynasties and royal blood as feudal Europe were. Of course there were dynasties and it was important, but there wasn't this ideas that you could only rule if you had imperial blood. If you became an emperor without any imperial ancestry, that wasn't a problem.

Keep in mind that Byzantines called themselves Romans... but no one else did. Feudal Europe didn't see them as the heirs to the Roman emperors, to them that was the pope and the Holy Roman Emperors. They just saw them as "the Greeks", who were also heretics. So no Europe didn't look for Imperial Roman blood after the fall of the Empire. There were some marriages with the imperial dynasties but they weren't seen really as anything special.