For a few centuries yes, but with a more consolidated Muslim world focusing on the big prizes, Rum and Constantinople, I could see the Byzantine Empire falling earlier.
This earlier decline of the Byzantine Empire, however, will paradoxically be good for the Carthaginian-empire-to-be, as it will increase the number of Byzantine scholars and nobility taking refuge in Carthage, and the Exarchate's navy will become the most important navy in the Mediterranean.
Really? Hmm, I can see the rationale, but I think it might be just as possible that the Exarchate keeps the Empire stable. Being Roman, and obsensibly part of the Empire, gives them some legitimacy to say - "no Anatolia, stay with the Emperor, good boy", and essentially support good sitting Emperors, whilst being able to effectively force a peaceful transition of power. Basically they could be an Emperor-Maker, which if they choose to go for it themselves, like Heraclius, could well win at the right time. So it could provide a measure of political stability as well as an economic and military partner.
The latter is where I think it provides longevity. The Exarchate has a pretty solid motivation for a strong, but not too-strong Empire. If the Empire in the East falls, its best ally falls, as does its trade partner. However it doesn't want an Empire with both the Levant and Egypt, otherwise it gets sidelined. When there are fractures in Muslim unity, I honestly think the Exarchate might ally with the Empire to set up Exarchs in the Levant and Egypt - not direct rule, I don't think it would support that.
Just taking advantage of a weakened Egyptian power, as the first crusade showed, but with a fleet from Roman Africa rather than the madness of the First Crusade, seems to be a sensible geo-political move. Setting up an Exarch in Egypt helps strengthen the Roman World, without directly strengthening Constantinople, and ensures that Africa is safe in the East, as now that Exarch is going to be trying to ensure an Exarch in the Levant, or to be the Levantine Exarch - freeing Africa to do its own thing.
At some point I might have to draft a 'Roman Commonwealth' TL based on this premise. I keep coming back to the idea.