Well, IOTL, the Exarchate DID break off, before paying tribute during the first (of 3!) invasions. My personal PoD is that rather than declare independence, the Exarchate needs to unify more Berbers around it earlier. It might mean more Berberisation (which isn't a problem), with the end result being the creation of a tighter, stronger military force that can stop the invasion - if the invasion was ~50,000, a force of 60-70k should be enough to stop and then counterattack to retake Tripolitania.
During that defensive campaign, they need to contact the Empire to have assistance, for a counterattack. A combined Romano-Berber invasion from the west, with full-force naval support from both the Empire and Exarchate fleets could well be enough to take advantage of the rebellions in Egypt during Uthmans later reign to recapture Egypt.
That shifts the burden of being the "Flank" in the Romans war for existence against the Caliphate to Egypt - ensuring that Africa only has to hold out against Spain.
At which point, holding the region seems more than possible to me, it just relies on an accord between the Romans and Berbers. Perhaps a settlement that the Exarch isn't an Imperial appointee (or just an appointee), but elected in some way by representatives of each city and each loyal tribe. Either that or they select a pool of candidates for the Emperor to choose. Either way it is some resistance to Constantinople in terms of power over choice, but if gives the Berbers a stake in the Exarchate, which could be enough to reconquer the Atlas Mountains and stabilise the Exarchate as a long-term force. After which, in a twist of history, puts the Exarchate in position to dominate the Western Med by sea. That might lead to rampant piracy, or complete control over the sea lanes (which... there is a fine argument to if that is different).
Whether it breaks away or not comes down to how onerous Constantinople is. Berber forces to reinforce Egypt can only happen so much, and supporting invasion of Italy can only happen so much. Asking for them to retake Spain - no deal, at best a support fleet/protecting logistics.
So yeah, you need the Exarchate to not be the front-line in two warzones, it needs the Berbers, and a hands-off approach. At which point it isn't the "Breadbasket of Rome", but the "Western Fortress"