When During Time trial he sets a new track record racing with my previous ghost. When I start a new lap the ghost doesn't update and every time i want to race with my best time I have to restart my time trial to update the ghost.
As one of your ghost, use 'session best'. The first lap there is no own ghost. With second lap you will have the ghost from first lap. If you then set a new personal best, your own ghost is updated to that. When starting TT, using his own best ghost which then gets updated doesn't work AFAIK. But you can set one ghost to your best (till that time) and a second with your session best.