I'm afraid I'll have to disappoint you here, I have never heard of a Jewish-Sassanian Commonwealth before. A quick look at the the wikipedia page reveals that it was mostly based on this article by a Rabbi Ben Abrahamson, who is apparently a proponent of linking Islam with Judaism. There is nothing wrong with that claim itself, there are plenty of evidence suggesting this, but his website does look a bit dodgy and at the very least, he is not an academic historian and his article makes leap of logic that I don't agree with.

The two main sources covering this pivotal period, pseudo-Sebeos' Armenian History and Antiochus Strategos' account of the Sack of Jerusalem, do mention that Jewish unrest played a significant role in the fall of Palestine, but it is clear from both sources that it was Persian officials who actually governed the province once it fell. Then there was a civil revolt in Jerusalem against the Persians (which included a pogrom of the local Jews), resulting in a Persian reprisal and the sack of the city in 614. However conditions for the Christians quickly improved in the following years according to pseudo-Seboes - he included a letter from the temporary Patriarch of Jerusalem asking the Armenian Church for aid in rebuilding the churches of Jerusalem, date-able to 616/7 at the latest. In the letter, the Patriarch blamed the Jews instead of the Persians for setting fire to buildings in Jerusalem (no doubt because there were Persian officials looking through his letters) and noted that Jews were banned from moving into the city. Evidently, the status quo survived.

There was another source cited by the articles describing the Jewish-Sassanian Commonwealth, the Apocalypse of Zerubbabel. I have never come across this before and the historians of the seventh century I've read have never used it. According to this, there is doubt that it was from the seventh century at all, but may only be from eleventh, which together with its religious nature, makes it a questionable source for history, especially when we have the reliable work of pseudo-Sebeos to compare it with.

Basically, there is no indication that Jews were placed in charge. They were freed from Christian persecution yes, and no doubt some were rewarded with positions of power, but government remained in the hand of the Persians. There was no reason for the Persians to hand over power to the Jews when the Christians were the overwhelming majority. Moreover, there is some evidence that the Persians were seeking to integrate the Levantine provinces and Egypt into its empire, which meant imposing the formidable bureaucracy created by Khusro II on conquered territories, rather than creating autonomous provinces. Byzantine control was weak when their forces returned to the region after 628 (not 10 years before the Islamic invasion as you said, only 4 years actually), but that had nothing to do with the creation of an autonomous Jewish province. 26 years of total war, combined with a generation of Near Eastern inhabitants who had never known Byzantine rule, was more than enough to weaken the Byzantine frontier with Arabia.

Edit: Should have also added that during the war with Persia, Byzantium lost all of the Near East and Egypt, suffered many successful raids into Anatolia, had its Balkan territories almost entirely occupied by Slavs and its capital besieged for the very first time in 626, not to mention the end of the Byzantine enclave in Spain in 624 and the revolt in Italy in 619. Jewish unrest, no matter how major, was a minor issue when compared to all the other crises faced by Heraclius.

Sources:

The Armenian History Attributed to Sebeos, translated by R Thompson and Commentary by J Howard-Johnston, 1999

The Roman Eastern Frontier and the Persian Wars: AD 363-630 by S Lieu and G Greatrex, 2005

'The Persians in the Roman Near East (602–630 AD)' by C Foss in JRAS 2003