Persian Satraps ! Also Alexander's Diadochi system !
This is a really great option to be honest, especially if we take Alexander and have him survive his poisoning/malaria as a much weaker man physically, forcing him from this young conquerer to a prematurely old ruler. Have him rule long enough for someone like Hephestion to be undisputed as his primary general, and then die with a child-heir. That creates a perfect situation for Hephestion to start the whole process, he's been appointed to the position by Alexander, and confirmed by his son, who can effectively be put into a gilded cage in Babylon whilst his Commander-in-Chief is in charge.
Alternatively, he's not in Babylon, but instead, Pella - ceremonially important to the Macedonians sure, but not the best place to rule the Argead Empire, much like Edo was better for the Shogun, maybe Antioch or somewhere in N.Mesopotamia does it for this empire.
Do I think it'd always be this way? No - I expect you'd see at least one attempt at Imperial Restoration, and probably more regular shuffling of which family was the CiC.
Though, as much as I love the idea, I'm not sure it could be as (heh), unified, as the Japanese Shogunates. United over an island chain is much easier than across the Persian Empire and then some more (are we really pretending Alexander would stop conquering even if he didn't get to it himself?). In which case we might get something more fragmented, with divisions within a theoretical whole. Ptolemy confirmed as the CiC in Egypt, another in Bactria, another in Persia. I don't see it as impossible as this forming an institution where the Emperor of Macedon/Argead Emperor is given similar reverence, but not the same secular power. Main source of legitimacy for fighting other parts of the Empire, significantly powerful locally (i.e. King of Macedonia, Archon of the Hellenic League
and Lord of Asia, etc), but not able to exert authority over any of the regions without their true leaders consent. (i.e. might be able to found a city in Bactria, but it's a cold day in hell that it'd pay taxes to Pella over Bactra.)
It might also not happen as quick as proposed with "Just After Alexanders Death" - regionalisation might be slower, but without the sudden collapse of Persia, followed by the sudden death of Alexander? You've got the potential for this Empire to become a similar entity (though I'm not sure it'd be called the Persian Empire if it grows further west. Macedonian Empire? Empire of Asia-Europa?)