Well, it certainly could have existed longer than IOTL, but it's less related to geopolitical situation than inner policies.
One of its big problem, at least militarily (but it covers some more broad policies issues) was the ethnicisation of Fatimid army : you had quite distinct groups, mainly Berbers, Sudanese and Turkic (with smaller ones, as Saqaliba, which may more mean Caucasian or non-Islamized Turks in this context).
Giving that military position meant political domination, you see it had the potential to get ugly really, really quickly.
And, eventually at the first opportunity, these different groups fought each other, causing much trouble. More important the crisis, more important the mess.
Even if the situation was more or less resolved (using a Saqaliba general), after the second part of the XIth century, the power definitely went to military leaders, and the consequences of regular infighting lasted quite long, without even mentioning the more or less related loss of North Africa and huge parts of Syria.
While Norman and Seljuk conquests are hardly butterfliable by one PoD (both being largely unrelated), having a stronger Fatimid Egypt at this point would have only helped : unfortunatly, the nature of Fatimid rule makes it hard.
Maybe, but that's a large maybe for a large PoD, butteflying away Seljuk and Turkisation of armies in Western and Central Arabo-Islamic world would help.
If, you could manage to bind that (or using it as an earlier PoD) with butterflying most repressive decisions of al-Hakim, it would be perfect.
Not that, in spite of his reputation, these were insane or fanatical, but they were erratic, and caused some political-religious trouble, within and outside the Caliphate.
Don't get me wrong, you'd still have important crises after his death, but without organized Turkic armies, with a more tolerent religious policy (which could quickly go against the wishes of the population, apparently annoyed with non-Islamic populations' prosperity), it would only help making these crises less long-term impacting.