The President apparently asked for options but his military kept presenting him with the same ruinous proposal dressed up in a thousand decorative variants: prop up the recalcitrant puppet regime and smash down hard on the insurgents. Along with everyone else the President was unconvinced but felt he had no other choice. And so the hopeless slide towards defeat continued.
At times like these we ask ourselves "What would the Culture do?". The Culture (prop. Mr Iain M. Banks) is a proxy here for a civilization of supposedly enlightened values with essentially limitless power. It's a way of asking whether even an America without resource constraints could prevail.
Option 1: If you've got it, use it.
The Culture has weapons technology to destroy stars: it would be a matter of milliseconds for the country concerned to be converted to lunar regolith. However, America has the nuclear capability to do that today as well and chooses not to do so. Why is that?
The Romans had a pretty good empire for hundreds of years. This is how they dealt with rebellious Carthage. "The Romans pulled the Phoenician warships out into the harbour and burned them before the city, and went from house to house, capturing, raping and enslaving the people. Fifty thousand Carthaginians were sold into slavery. The city was set ablaze, and in this way was razed with only ruins and rubble to field the aftermath." Apparently they didn't actually sow the fields with salt, but perhaps they didn't need to. [Wikipedia].
This level of ferocious ultra-violence works but only in a unipolar world where everyone else just gets more terrified afterwards. OK for the Romans, not OK in today's world.
Option 2. Hearts and Minds (involuntary)
The Culture would certainly be able to interfere with the "terrorists"' minds - twisting them so that they couldn't or wouldn't fight any more. This level of technology is beyond that available to America. However, targeted assassination is a less subtle variant and is already a minor feature of the current conflict.
The Culture would have a problem with this tactic though because it's unethical. It makes zombies, puppets of human beings. It’s not a solution.
Option 3: Heart and Minds (voluntary)
What the Culture would actually do (cf. Inversions, The Player of Games) is to infiltrate their society with Culture agents and try to steer the protagonists to a more enlightened world-view. The Culture does not believe in moral relativism, does not believe that mediaevalist societies with oppressive laws and beliefs are actually a good, tolerable or even valid option. So it would gird itself for the long haul, the long march to cultural transformation. The end of that journey is a non-oppressive society, not necessarily US-style bourgeois democracy.
America hasn't really tried that, but then again, it isn't the Culture. And it does have a short-term problem.
Note: I’m aware that I haven’t answered the question of what America should actually do. I think given that the West is not going to concede the Caliphate, Joe Biden probably has the nearest thing to a workable way forward.