The Basileus achievement doesn't check for the "has_switched_nation = no" in happened. So if you want you can escape by annexing a nation and play as that nation (you have to start as Byzantium though).
Then win your independence war easily (since the Ottomans will attack Byzantium as well), convert to orthodox at some point and get Greek or Pontic culture. Then conquer the remaining provinces and the achievement is yours.
Check the reddit thread about it for more details:
https://www.reddit.com/r/eu4/comments/76c5aq/tip_basileus_too_hard_as_byzantium_you_can_tag/
Then win your independence war easily (since the Ottomans will attack Byzantium as well), convert to orthodox at some point and get Greek or Pontic culture. Then conquer the remaining provinces and the achievement is yours.
Check the reddit thread about it for more details:
https://www.reddit.com/r/eu4/comments/76c5aq/tip_basileus_too_hard_as_byzantium_you_can_tag/