"...being an Islamic extremist."
I'm not sure if I've read this correctly, but I thought that they knew that he was a Muslim but not that he was an extremist. I'm not American but I don't think that it has yet been made illegal to follow any specific religion - nor should that ever be the case! An individual's religious beliefs has nothing to do with the State. Many of those close to him DID know that he held extremist views but, apparently, they did not take the necessary steps of raising the matter with anyone who could assess his suitability for either his post or for buying firearms. There were lots of mistakes made but I don't think that the FBI were to blame for them. It wasn't the case that the FBI 'had no problem' with the person that you describe - rather, they we not aware of the facts because nobody bothered to tell them.
Not being an American, I might have missed some critical reporting but that's how I recall it being reported here.