Were Heraclius' sons men of stature or particular intelligence?
Constantine III seems to have been- he was certainly popular, but, as you say, he took ill and died a few months after his father, leaving an uneasy tripartite regime made up of his stepmother (and cousin!) Martina, Martina's eldest son Heraclius II and his son Constans II. The former two were deeply unpopular because of the incestuous nature of the marriage between Martina and the elder Heraclius, and were soon deposed by an Armenian general named Valentinian, with Constans II becoming Emperor. That Constans was a precocious child, twelve years old upon taking the throne, who was a decisive and determined man throughout his reign, and who won several victories over the Arabs, ultimately forcing them to pay him tribute. He also campaigned with some success against the Sclavenes.
Now, assuming a 630s POD, this could very easily go differently. There are plenty of the Heraclian line around, but only Constantine III and his children are especially popular. Martina and her children will be safe for as long as Heraclius is alive (the marriage was apparently a very close one), but afterward, I think she'll definitely struggle to hold support for very long.
The best case scenario for the Empire, IMHO, is to have Constantine III succeed his father late in the 630s, avoiding Heraclius' descent into madness and ending his reign on a high. Constantine could be a popular and inoffensive caretaker Emperor for a decade or so, and eventually pass the throne on to his son who will be several years older than he was when he took the throne IOTL.
That said, I suppose a best case scenario isn't necessarily the one most interesting to write about...