My knee jerk reaction was to think "how stupid that someone investigated by the FBI got a gun legally".
Having had time to think about it, I can't possibly agree that this is a good idea for many reasons.
There are no objective criteria for being on the list. It is likely the criteria would be changed, so that more people could be put on the list, since many will react with outrage if anybody displaying vague hints of trouble was not on the list. I could envision the FBI simply putting all Muslims on the list. Anyone observed going to a mosque.
It would also add fuel to the roaring fire of more surveillance, less privacy, and abuse of national security letters. "Well, we had to get all those phone records so we'd know who to put on the no-gun list."
Without due process, I can't abide taking away a right. It's bad enough when it is used to revoke a privilege like air travel.