Hm. Noone teaches Byzantine Greek as something distinct from Ancient Greek. That’s because for most purposes, it isn’t distinct.
I’m going to go through a potted history of Byzantine Greek for others who might stumble on this question.
There are three registers of Mediaeval Greek to consider; I’ll use Mediaeval to include Greek under Latin rule.
- The vernacular doesn’t show up much at all; nothing systematic before the 14th century. There is exceptionally a vernacular corpus from the 9th century, the Category:Bulgarian Greek inscriptions. You don’t need Modern Greek to read them.
- Low literary Greek was an officialese Koine, with occasional hints of vernacular developments, and lots of Latinisms.
- High literary Greek was Atticist: it was an attempt to write in the Attic of the ancients, with varying degrees of over-enthusiasm.
If you’re going to work with Mediaeval or Byzantine Greek, you do the following:
- Learn Ancient Greek. There will be atticisms of various flavours, a bit more than in the Gospels (though some church fathers had reasonable Attic learning).
- Be across Koine.
- You won’t need to learn Modern Greek unless you are actually working on vernacular texts.
- Read a couple of histories of Greek, so you know in broad terms what’s likely to come up. Medieval and Modern Greek (9780521299787): Robert Browning, A History of the Language and its Speakers (9781118785157): Geoffrey Horrocks
- There isn’t much in terms of a grammar of Byzantine Greek; An historical Greek grammar chiefly of the Attic dialect [By] Antonius N. Jannaris is still the best out there. There are specialist articles, but you likely won’t need them.
- Bring a dictionary. Bring three, actually: A Patristic Greek Lexicon: G. W. H. Lampe; Lexikon zur byzantinischen Gräzität; Greek lexicon of the Roman and Byzantine periods (from B.C. 146 to A.D. 1100) : Sophocles, E. A.
- Be prepared for oddities. The oddities will depend on who’s writing and when. Atticists get enthusiastic to the point of fictional grammar. (When Harry Turtledove was a Byzantinist, he wrote “Most Byzantine historians felt they knew enough to use the optatives correctly; some of them were right”.) Byzantine authors systematically misaccented words, because they could. Some authors slip up and let vernacularisms in, though that’s more a late Byzantine thing.