Certainly not Byzantine Empire - even Byzantines didn't call themselves that. As far as "Romania" - it's technically possible, but extremely doubtful - there's no real continuity with the Eastern Roman Empire (which never positioned itself as a Greek state in any form), and "Megali Idea" wasn't so much about restoring that state as uniting all Greeks under one country. They didn't have any claims on Egypt or Ravenna, for example. Also - the Byzantine heritage wasn't valued as highly in late 19th century Greece as Ancient heritage was - for example the language of the period was considered corrupt

So unless the King or government at the time were extreme Byzantinophiles - I don't really see that as likely to happen.