Minor correction, Artaxerxes III was III in Egypt as well (and thus IV was IV in both as well).

Though Egypt was independent for most of Artaxerxes II's reign, he was still acknowledged at Elephantine, and likely much of southern/central Egypt, as late as 401 BCE. Given that Amyrtaios was not acknowledged in the south until that point, it's most likely that he only seized the whole country when the Persians were distracted by Cyrus the Younger's revolt. Less directly stated, this shift can be plausibly connected to Xenophon's account of Abracomas, Satrap of Syria (Assyria/Eber-Nari) in the Anabasis, who was summoned east with his army to aid Artaxerxes II in 401, effectively removing the pool of Persian reinforcements closest to Egypt.

Technically, we also don't know if that coin represents Artaxerxes IV. It's just a best guess based on the apparent age of the figure depicted. It could represent Artaxerxes III in a variation of the beardless style common in Late Period Pharaonic art or one of Artaxerxes III's other sons who were killed to pave the way for Artaxerxes IV.