To sum it up:
Latest "Easy" POD: Before 1204.
Latest "Hard, but not entirely wanked" POD: 1340(before the Civil War).
Latest "Wanked to the extreme, but maybe sorta plausible" POD: 1402
You'd need to have Timur really smash the Ottomans a lot more, have a warrior emperor show up, maybe have a successful crusade take place, and have the stars align for Byzantium... and it could possibly survive in Greece as a minor statelet.
1453... there's nothing left, no enemies that could turn back the Ottomans, and even if, somehow, they were... the remnants would get nabbed by someone else anyway. The only curious possibility would be if the Ottoman Empire fragmented really terribly for some reason, and the Morea emerged as a petty state (probably under the influence of one or more Italian city-states). Constantinople is screwed, it's too much of a symbol, and too weak to survive.