The best type of anesthesia depends on the patient, procedure and surgeon. Please see a board certified plastic surgeon for in person evaluation.
Fat transfers can be done awake, however, the Lidocaine volume used to numb might limit the amount of fat transferred. Also, with local anesthesia, there is a limitation of the amount of local anesthetic that can be used. If you want several areas treated with liposuction, you might want to choose general anesthesia.