In the case of Lee Harvey Oswald – it seems that the Oswald name with variations – “Lee”, “Harvey”, etc. was used by multiple undercover aliases at the same time, and in 1963, one of those aliases was taking directions from both CIA and FBI undercover. Insight into the way the CIA schemes leads me to believe that setup blame for the crime on a guy who taking FBI undercover orders would have been seen as one factor helping to guarantee FBI cooperation with the coverup.
It’s a big reach…but I can imagine, by a similar logic, that exposing Anna et al. to scandals that trace back to both CIA and FBI helped guarantee they would both fight hard for continued overup.