As a Coalition faction, these guys didn't make sense because of their religious nature that the Coalition WOULD suppress. The Coalition wouldn't allow a Caliphate or any sort of expression of Muslim identity to flourish. Arab, maybe, but never Muslim, even if it's faux-Muslim and revolving around some Starlancer ace.
Which leads me to another point: the Coalition wouldn't allow these guys to detach from the Coalition, but would allow some degree of expression. Maybe there's an Arab-majority fleet of the Coalition, a CPW-Medina somewhere, and maybe Exeter is to be reclaimed for the Coalition's Arab minority as planet idk Damascus.
If you want to shape an Arab identity among these people and as part of the Coalition, I'd rather take example from the Middle East's secular or "secular" republics, for better or worse, Syria, Iraq, Egypt, Yemen.
A Caliphate within the Coalition's midst would be a huge nono for the government.
Now, on to the second revision, which turns them into Corsairs.
There could've been Arabs on the Hispania who would've just said "Fuck the Coalition, let's live among these guys in this desertic hellscape with constant famines", but if the premise that they're Arabs is based on the idea that they discovered last thursday that they were Arabs and engaged in some sort of cultural reconstructionism, that's a wrong premise.
Even more so when they have the pointlessly grandiose objective of being a House.
The Custodi pulled off the minority-within-the-Corsairs thing excellently. They were Greek, they had a distinct culture from the planet's Hispanic majority, they settled in their own area which happened to be the one fertile area of Crete, turning them into that world's peasantry, they went as far as naming Crete's moon, and never had any secessionist intention because of how integrated they were in Crete's fabric, they just looked out for themselves.
My feedback if you want to keep them Corsair:
they kept their culture throughout the centuries (didn't see that coming, did you? It may sound counter-intuitive, but if the Custodi did that, so can these guys), settling en masse in their own region of Crete. Because of Crete's decentralized nature, they had their own political autonomy, it could be a Nasserist republic, either kind of Baathism, some sort of Islamic emirate or sultanate, up to you. They're Coalition stock, so they're probably a highly secularized society and I see them more sticking to the stereotype of the broken Middle Eastern oil-exporting republican dictatorship, but that's up to you. Maybe they even supplied fossil fuels to Corsair ships, what do we know? Maybe they have a friendly rivalry with the Greeks over who gave a name to Tripoli, after all there's Libyan and Lebanese Tripoli and a Greek one in Morea. These kinds of neat touches.
And no grandiose aspirations of Housedom, realistically they'd just advance their interests and, when that's possible, Crete's as a whole.
Problem is, if the Hispanics find out they're Arabs and hence Coalition agents, they'd kill them horribly for having sabotaged their sleepership. They WOULDN'T make it eight centuries, they'd be murdered before landing on the planet.
So yeah, Battlegroup -some Arab name- as part of the Coalition's fleet makes far more sense.