Chowchilla Women's Prison, where Lorelei was being kept, is a California state facility. It makes no sense for the FBI to divert her to federal custody and then put her in a California prison, where state authorities like the CBI would be able to question her.
When Jane first pulls into the parking lot to stage his kidnapping, the shadows are long, indicating that it is early in the morning. But then when he goes to smash the car window, the shadows are much shorter, indicating it is now later in the day closer to noon time.
When Lorelei and Jane are at the beach, water droplets can be seen on the camera lens.
Jane uses his own tire iron to smash his car window and puts it in his trunk, leaving his own fingerprints on it and nobody else's, which calls into question whether somebody else used it to smash the window.
A seasoned investigator would quickly notice glass on the driver's seat of Jane's car and conclude that it was staged. If the window was broken while Jane was sitting in the car, the glass would end up on his lap.