Terrorists, Mafia and KGB
For the war against terrorism to be effective, it is necessary first to know how terrorists operate. And what one finds out is that they do not operate like the military, but rather like organized crime. They are sneaky and underhanded. They present themselves to their communities as their saviors and use community loyalties to protect themselves. They are a lot like the Italian Mafia, and to defeat them it makes sense to draw on resources that have had success in fighting organized crime.
So it makes sense to draw on the knowledge of FBI, but more importantly still on the Russian spy agencies. The KGB was the most effective organization of its kind. And while talk of KGB brings back memories of the Cold War, on this matter at least the Russians are allied with America. The Russians no more want to see terrorists killing their people than Americans want to see terrorists killing theirs.
So my advice would be to work with the Russians on this and draw on the knowledge that they've amassed, both during the Cold War and during the Putin period. And while much of that knowledge was used for wrong, that does not mean that it cannot be also used for more rightful purposes, any more than the fact that guns can be used to murder means that they can't also be used for self-defense. My advice to the counter-terrorists therefore is to work with the FBI as well as with Russian spy agencies to draw on the knowledge they've amassed fighting organized crime and internal enemies and use that knowledge for the common purpose of defeating the terrorists, whose modus operandi is very similar to that of the groups that these agencies learned how to fight.