I feel like the period of the Bronze Age Collapse (1200 - 900 BCE) would make sense as the genesis point of a new Aramaean religion (there was a failed attempt to convert the Egyptian national faith to monotheism just over a century before, and of course Zoroastrianism started somewhere in the mid- to late period of the second millennium BCE), but beyond that I've got nothing on how or why it would be "Islam-like." Presumably a monotheistic faith arising at that time would be derived from one of the established Mesopotamian deities, but which one I've no idea.
TBH I think you may need to figure out what work certain terms are doing, first. Like:
What does "Islam-like" mean, exactly? In what ways is the religion "Islam-like," and why? (Islam had specific precedents to work with, and was born from specific internal and external pressures. Do the Aramaeans have anything similar? If so, why don't those pressures just lead to their using their OTL religion to fill their needs and as a basis of empire as the Assyrians did?)
What makes a "Caliphate-like" state, and why should the Aramaeans prefer it to a system like the OTL Neo-Assyrian Empire? (Does "Caliphate-like" mean the king also being "commander of the faithful" in some way? What would that role look like and how would it differ from the religious roles and duties already inherent in being a Mesopotamian king/emperor? Would it essentially be a return to the earlier paradigm of Mesopotamian priest-kings, and if so, how is the "Caliphate-like" state addressing the reasons those roles originally split off from each other?)