Just handle official business by the book, so really, the people from the FBI weren't brought in by Jane, because it wasn't necessary.
As some situations develop to a certain point, it doesn't matter who handles it, so whether it's an insider or not, the effect is pretty much the same.
At Simmons's level, it doesn't matter who investigates, because in the end, after all the probing, Jason is bound to be labeled insane.
When it comes to assassination, the FBI is more professional than the CIA, by a large margin at that. Their processing methods are formulaic routines that can be directly applied.
Even dealing with Gao Guang and the others followed a procedure: just on-site questioning, merely a conversation right there, nothing that involved taking them to the station for clarification.
These are people with faces and clout, with wealth and power, how could they ever be treated as suspects?