“If the cameras on public property, I guess the government, including the Navy, has a bit of a say over whether it shows things.”
Nah, legally, anything on public property, or easily visible from public property, means there is no expectation of privacy. That’s why people can record you with their cell phone when you are out in public and saying “I didn’t give you permission to film” is irrelevant.
The cameras were permanently mounted on NPS administered property. NPS is under no obligation to extend permission for fixed-mounted cameras in their parks.
A good lawyer might argue that past history constitutes a right by waiver but then you’re up against the Gubmint.