This would be highly out of character for the U.S. government. The U.S. Government has done its fair share of unseemly acts, but completely making up an enemy and attacking them as a cover up is something it has never done. If there is some event like this that I am not aware of it either happened a very long time ago or was not done on this kind of scale.
And there is a reason it has not been done on this kind of scale, if ever. Governments cannot cover up things on this kind of scale, and blaming a foreign power or terrorist group would be so easily falsified that it would be completely pointless to try. Not only would be incredibly easy to falsify, but there would be too many layers of government involved for this to even be practical. Something like a CIA coup in the 1950's or 1960's was possible without anyone knowing because the entire operation was clandestine and could be authorized and carried out without having to involve anyone outside of the executive branch. This wont work with the battle of Chicago because it is a massive event and everyone will be involved. Congress, state governments, local governments etc.
I'm pretty sure they've done it since WW2 but I don't feel like looking it up right now. It's true that real false flag attacks, by any government, aren't on this scale.
However, the lie could go farther than you think. After all, they didn't invent an attack. If you line up enough "experts" to go out there and say "It was the Iranians!" enough people will fall for it because they will want to believe that there's something that can be done to stop it from happening again. The alternative is for the President to go on record saying "there is no effective action we can take to stop this from happening again." That's just not going to happen.
Will the truth come out? Sure. But the only people who have anything close to a complete and accurate picture of what happened aren't talking because they're the ones who decided to cover it up. By the time there's clear and convincing evidence it wasn't Iran, or whoever, the consequences of the lie will be manifest and there will be no going back.