Here in Florida, lighting protection is only REQUIRED on (3) occupancy types; Hospitals, Nursing Homes, & Educational Facilities. A few counties have amended their local code of ordinances to include all public owned buildings & emergency shelters. In these cases, permitting and inspection would be required.
So technically, if you were to install an LPS on a structure that is not required by code or law to have an LPS, no permit would be required and thus no inspections.
The craziest part of the whole matter is that LPS installers are exempt from ANY type of state license. A few local jurisdictions have competency & occupational licensing requirements, but not many. And many of the jurisdictions that do require permitting of LPS's, only require submittal of 3rd party (UL or LPI) certification after the work has been completed and perform no additional or filed inspections.