Plausible it isn't, but it sure would be fun. I had the Roman Empire go Jewish once...
Now, if you allow for the Jewish faith to be much more accomodating to converts (and at certain times it was, so it isnt that outlandish) the question is, would it acquire the same missionary drive as Islam and Christianity, or would it stay welcoming, but non-mandatory? Personally I don't see how Judaism allows for a missionary imperative, but history has seen weirder things.
Without this drive, persecutions would be unlikely outside the faith. Within the flock, on the other hand, things could easily get quite nasty (remember King Josiah?). Assuming there will be a Jesus in this TL he could be crucified (the Persians knew how), though I think it more likely he'd suffer a traditional Jewish stoning. After all, if the Persian Empire is Jewish, the Jewish authorities won't be officially barred from carrying out executions. His followers, too, would most likely have a hard time because they'd be seen as straying Jews, not gentiles who'll believe any old thing. I wouldn't be surprised, though, if the altered structure of Judaism as a dominant imperial faith (in the context of the generally tolerant and easygoing Persian empire) wouldn't have a niche for - Jesuanic Judaism? If the huge weight of persecution, fear and inferiority complexes is taken off the faith I would expect a very pluralistic and tolerant atmosphere to be possible.
You'd have to ask yourself, though, whether Jerusalem could retain its dominant position. Holiest place, yes, but would religious dogma (as far as it goes) not rather be hammered out in Persepolis or Susa, or Babylon? Alternatively, if the Persian kings actually make Jerusalem their residence (I'm told it's really nice in spring), that would entail a shift westwards for the Empire. So much for evletheria and avtonomia...