I think that it would naturally be Romance speaking. After all, it would be isolated from other Muslim territories, and if the Shia rulers remain considered as enemies by their Sunni successors in North Africa maybe they could promote the use of the local language to create a kind of "national" identity against the followers of the "false" Caliph. What about this idea?
I really don't think national identity matters in any way (until around 1000 years after the time period we are talking about), so isn't relevant. The Fatimid founders were Arabs, they spoke Arabic, and most of their supporters were Berbers who spoke various Berber languages. Southern Italy was cosmopolitan as well: the Lombards spoke various Italian dialects, Greek was spoken, as was Arabic.
Furthermore, I don't think the concept of "natural identity" even
existed that long ago, nor do I think was it needed. The Fatimids would not be entirely isolated, as they would retain influence throughout North Africa. Who's to say that the new caliphate would not be Ismaili Shiite as well? One never knows.
But with regards to the language, I do not think the situation would change drastically. The Lombards would continue to speak Italian, although as increasing numbers converted these Al-Ankubarti would learn Arabic as well. The Fatimid remnant bureaucracy would use primarily Arabic. Berber settlement would be reasonably high, introducing that language into the mixer. But over time, as the Al-Ankubarti come to dominate Fatimid politics, there is a possibility of an Italian version of Mozarabic gaining influence. However, Italian Mozarabic might just be restricted to Italo-Lombard Christians living under the Fatimids.
If this shadow-caliphate survives long enough, maybe the Fatimids would fall to an Al-Ankubarti revolt. But that would be centuries off, so I can't begin to speculate.