Who do you imagine would be raising Alex IV ? His father and most of his paternal relatives were dead but his maternal relatives, including a very powerful great grandmother, Sisygambis, are still alive. Most of the generals would be too busy fighting each other than focus on a baby's education, even if it's a baby King.
If Alexander IV is to have an chance whatsoever of being Alexander IV - the people who raised the OTL baby.
No one is going to let the heir to the throne of Macedon become a barbarian (by the Hellenocentric definition of the term) if they don't just throw him in a ditch.
You can't have it both ways. Either the generals act like they do OTL, in which case he'll be lucky to see adulthood, or they don't, in which case he'll be very definitely part of the foreign, Greco-Macedonian rulers that the Persians would rather not follow given a choice. Raised by his mother and her family isn't an option and isn't even desired by the Persians.
As for independence, I imagine the Persian Satraps were no fools. If they where to revolt it would present a major threat to the squabbling generals. If the generals ignore the revolt it would invite the other native peoples to follow their lead. None of the generals would want to be remembered as the man or men who lost Alexander's empire.
Didn't stop them from doing what they did OTL. Didn't make the satraps loyal OTL.
Its not even a matter of open revolt. Its a matter of "By the way, while you guys fight, we're ignoring all of you,"
As for the Lancastrian reference, look at the Angevin empire and Plantagenet dynasty. The Angevin's were french, spent more time in France than England, and controlled more French land than the French king. Under John of England nearly all the land in France was lost, but John didn't lose the English throne. Sure when he died a french prince had been invited to take the English throne, but after John died leaving an underage son mind you, the nobles supported Henry III (John's son) over a capable adult. So it is possible to lose your homeland and keep your throne.
Laying aside an argument on how John didn't lose the majority of the French lands because Aquitaine dwarfs the size of the area he lost - this is not similar. This would be as if a foreign conqueror (which John is not - yes, the Angevins are French. So is most of the "English" nobility. Nor is he a conqueror or even son of a conqueror - Henry II was acknowledged heir to the English throne after the usurper Stephen by pens, not swords) lost his homeland and then tried to keep the lands he conquered, which is why I used the Lancastrian idea.