This isn't a big conspiracy, which is no small part of what makes it plausible. It's not faking the moon landing, which would require the consent and then silence of thousands of people spread all over the country. It could have been just FDR by himself, or at most a few high-ranking officials in the right places.

As for the scale of the attack itself, maybe they were hoping for something smaller. It's like the FBI winding up right-wing cranks and playing fast and loose with goading them on. Maybe the idea is to get them incriminated enough to make a big splashy arrest, but that goes sideways sometimes.