That's an interesting comparison and worth some consideration. I'm not sure I'd go so far as to call the Kai the equivalent of an Iranian Supreme Leader, though -- if for no other reason than I see no evidence that the Kai can formally veto or order stopped the policies of the First Minister if the Kai disagrees with them.
Maybe it's more like the British monarchy in the late 1600s and 1700s -- an institution that's not truly politically neutral but where most of the power lies in the democratically elected parliament?
IIANM, the Supreme Leader doesn't have any delineated constitutional powers; what Iran's constitution does is broadly restrict what the political government can do in accordance with the interests of the Supreme Leader, without ever saying what the government can and cannot do. Nevertheless, all the tools of power--military, policing, taxation, policy, legislation, etc.--are in the hands of the government, not the Supreme Leader. What tools the Supreme Leader can use are largely his prestige and influence should he oppose measures of the government.
The British Monarchy had money, land, control of the military. It's power may have been circumscribed by the Parliament, but it had real political tools. Into the early 20th century the crown exercised strong and active powers in the broad range of diplomatic issues, which affected how Britain approached both World Wars.
The Supreme Leader is far from a perfect analogy. The Kai seems to be a religions leader only, and the First Minister has all the tools of the government. When the Dominion threatens the station, Kira offers to go the the Council of Ministers. However, it is the Kai who takes the initiative to negotiate with the Cardassians and who participates in the process of incorporating Bajor into the Federation. The episode "In the Cards" suggests that she could have approved a treaty with the Dominion on her own; consulting with the Council of Ministers was just punting the issue. There are plenty of people who have intellectual, cultural or religious influence, but who don't affect the policy of the state. Even the Popes, pre-Italian Unification, didn't influence states (except those under his direct control,of course). Perhaps the Judges of pre-monarchy Israelites might come closest to the Kai's power. Whatever they are supposed to be, understanding the Kai requires extrapolating from historical examples, as such examples are, at best, rare.
