Would the Byzantines have spoken Ancient Greek or something closer to modern Greek?

Modern Greek.

Being literate in Greek has always meant being literate in Ancient Greek; so all our evidence of the vernacular is tainted, right up until the Cretan Renaissance (and there it’s tainted in a different direction, of conventionalised dialect). In the period between the Arab conquest of Egypt (when the papyri run out) and the first experiments with vernacular poetry in the 12th century, we have almost no direct evidence at all, outside of Bulgar inscriptions presumably written on their behalf by Greek prisoners of war.

But what we do know and can reconstruct tells us that the spoken language looked close to Modern Greek by the 7th century, and the texts we have in the 12th century, though macaronic, are identifiably macaronic with Modern Greek.

There would have been registers of spoken language as with every language. We have a hint from Filelfo, writing in the 15th century, that the language of the court in Mistras and Constantinople was “purer” than everywhere else. That suggests a proto-Puristic Greek, with more influence from written, atticist Greek than in the countryside.

Can you create your own rules in conlangs?

What others said. Yes, but make sure there is an internal logic to your rule, and that you’re applying it consistently and meaningfully.

Klingon has an internally consistent story with its zero copula constructions: the pronouns in copula constructions (“he — teacher”, ghojwI’ ghaH) have been reanalysed as verbs, and take verb aspect endings (“he is being a teacher”, ghojwI’ ghaH-taH) and subjects (“Worf is a teacher” ghojwI’ ghaHtaH wo’rIv’e’, literally “teacher he-ing, Worf”.)

Suzette Haden Elgin once accused Marc Okrand of linguistic malpractice, because he’d said that Klingon pronouns have subjects. Think of all the kids whose understanding of grammar will be destroyed, she exclaimed.

Fool. Cairene Arabic does pretty much the same with its pronouns.

But the key is that the rule has to be internally consistent. As Jim Grossman says, the rule likely makes more sense if the noun denotes a nominalisation to begin with. And as Zeibura Kathau says, the rule as stated is probably not what a linguist would end up describing it as—they would talk of complements of nouns instead.

And be aware of what ambiguities and dysfunctions the rule could introduce. You can have the same particle for objects of verbs and complements of nouns, as Olivier Simon does. But what happens when you have both a verb and a noun taking objects in the sentence: it is clear which of the two the complement belongs to?