At most, the FBI knew Oswald was politically suspect and occasionally kept tabs on him. They had no basis until 11.22.63 to believe that Oswald was violent or might commit an act of violence. He was just one of thousands of similar loons that they monitored. The DPD apparently did not record interrogations at the time. That was not a standard practice. They started asking Oswald questions that they already knew the answers to see if he lied and to get him talking. It is revealing to see what he would admit and what he might lie about. You are making Mt. Everest out of a molehill.
The "tabs" they kept on him were poorly done, half-measures, off and on. I'm repeating myself but it seems to be necessary: Hosty, the agent assigned to keep track of the Oswalds, said he never knew where they lived from October 3 to October 30. Again: he didn't know where they were. On October 30th, he finds out they are living on weekends with the Paines. But he doesn't know where Oswald was living during the week.
Again, from October 3 to the day of the assassination the agent assigned to keep track of Oswald
never knew where he lived on weekdays and evenings. Even when Hosty was told about the Mexico City visit and the meeting at the Soviet Embassy with Kostikov
he doesn't interview Oswald. In his defense, he said his attention at that time was on watching the Birchers and Far Right, the Walker people. They were considered (rightly) a more serious threat.
The claim that "Oswald's every move was monitored by the FBI/CIA" is simply not supported by what we know. It didn't happen. I don't know how many times one has to say this.
In conspiracy world the FBI and CIA and "thuh government" are always on top of things, everyone is a brilliant on the ball agent, a cog in this smoothly running and controlled machine. In reality stupidity and laziness and sloppiness and bureaucratic inertia is how it really works. But that's what the CIA told me to write (ahem).