Do you have imaging that confirmed you have FAI? If not, I’d check that box first.
If confirmed, I think you might do well to take a closer look at surgical intervention. The impingement caused by the acetabulum can cause a great deal of damage to the cartilage and the hip labrum. Though soft tissue mobility work and improving motor control can help, if there’s a problem with the architecture of the joint, there’s no exercise that will fix that.