The FAA now allows the Aviation Medical Examiner to issue a medical certificate under the CACI (Conditions AMEs Can Issue) policy for kidney stones. You will need to provide to the aviation medical examiner the worksheet completed by your treating physician. If you do not meet the criteria for the CACI issuance, it is still possible to be issued a medical certificate under the special issuance procedure.
After initial certification by FAA staff doctors, subsequent renewals qualify for AME Assisted Special Issuance (AASI). This allows the medical examiner to issue an airman medical certificate after the FAA has conducted an initial review of the medical records.