So, i believe we should use this discussion feature. I would like to contribute to this mod in the form which i can, and that could be by imagining what could be going on in a scenario like the one discussed in real life.

(Note that i'm taking into account that the scenario in which this culture is probably going to appear is in Italy. If there is a eventuality in which a Frank dominates the Byzantine empire like in the 4rth crusade bookmark, it is more historicaly plausible that the dominators be absorbed in the local culture, so there should be no new culture spawning in Byzantium)

I point out that, as far as i know, the Greeks used to refer to themselves as "Rhomaiõn" or romans, that is because they believed to be the continuation of the Roman Empire. But it seems like the western world did not used the same name, i say this based on, for example, the work of Saint Thomas Aquinas "Contra errores graecorum" (Against the Errors of the Greeks) which is clear that the Saint calls them Greeks, instead of Romans.
It could be more historicaly plausible for the Greek population dominated by Frankish Rulers to be called Grec (Greek, in French).

There are scenarios which we could imagine happening in real life which can help guide us in the development of this mod. It could be that:

1 - Fraconization: the frankish rulers impose their language and customs upon the dominated population similar to the process of Arabization in the Arab World, where the rulers do not become absorbed in the local culture, like what happened with the germanic invaders of Italy, but the rulers effectly impose their culture on the dominated. The thing here is that the new culture should be in the LATIN culture group to be more historicaly plausible. The problem also is that if this is the case, than the name of the new culture should be Grec instead of Romaios. This represents a Westernization of the culture.

2 - Frankish rulers become Romans in their own language: the franks ruling over the Greeks (or Romans) start to become absorbed in that culture, but they retain some aspects of their previous culture, that's why it would be a new culture instead of the rulers culture converting to the old dominated culture. In this case, the name Romaios would be apropiated for the new culture, and for it to be historicaly plausible the new culture should be in the BIZANTINE culture group.

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If we want to dwell deeper in this line of thought we could try to answer this question: which of these 2 options would be more historicaly plausible?

We could look at what happened in real life to answer this question, but first let me make this clear: I'm no historian and I probably don't know what I'm talking about.

As far as i know there's some greek (or Griko) speaking comunities in Italy to this day, but the general pattern is that they became absorbed in the dominant power's culture, historicaly: italianized.

From wikipedia:

"Although many of the Greek inhabitants of Southern Italy were entirely Italianized during the Middle Ages (for example, Paestum was by the 4th century BC), pockets of Greek culture and language remained and survived into modernity partly because of continuous migration between southern Italy and the Greek mainland. One example is the Griko people, some of whom still maintain their Greek language and customs.

For example, Greeks re-entered the region in the 16th and 17th century in reaction to the conquest of the Peloponnese by the Ottoman Empire. Especially after the end of the Siege of Coron (1534), large numbers of Greeks took refuge in the areas of Calabria, Salento and Sicily. Greeks from Coroni, the so-called Coronians, were nobles, who brought with them substantial movable property. They were granted special privileges and tax exemptions.

Other Greeks who moved to Italy came from the Mani Peninsula of the Peloponnese. The Maniots were known for their proud military traditions and for their bloody vendettas, many of which still continue today. Another group of Maniot Greeks moved to Corsica."

So based on this i believe that the Franconization option is the one more plausible to happen in a frankish domination scenario.