Keeping the straits open could be done, by a President who really wanted to.
The first molestation of a ship in the straits would be answered by leveling the personal residences of Imadinnerjacket and each mullah in the ruling council. A statement would be made that any missile launched from Iranian territory would be answered by destroying something that the rulers would be upset to see destroyed. We would focus on the homes of the Basij, next.
To prevent this, they have put the anti-ship missiles into the control of fanatical muslim Iranian Guards, not regular military. One approach they could take (among many) would be to give the missile crews sealed orders: in the event of general attacks by the US military, they must stay hidden in their caves, and obey their sealed orders. Orders to the 100s of missile batteries might say, “remain concealed until day 20 (or 30, or 55) after the war begins, and then attack any shipping in the straits that presents itself.”
All they need are spotters with binoculars to watch the straits for tankers, and attack any that they see.
Fanatical Japanese, following their “last valid orders,” stayed in caves in the Pacific for decades in some cases. Iranian Guards, briefed to the above plan in advance, would have no problem hiding for months on end. They would not need a modern, functioning command and control structure to carry out their orders.
Like I said, the Iranians are not stupid, and they can anticipate our counter moves, and create a plan based on countering them in turn.
In the end, it might take entire divisions of US military on the ground, going from ravine to ravine and cave to cave, to root them all out.
If you have an idea of how to stop them from launching their mobile missiles (solid fueled, firing from near horizontal in minutes after emerging from their lairs) when we could not stop giant liquid-fueled SCUDs from fring in the open Iraqi desert, please let us all know.