The only reference to this on Bungie's web site is is : https://www.bungie.net/en/Help/Article/11896 When you shut down your PS4, did you unplug it and then press the start button (which bleeds off any residual power) a couple of times? That seems to be the main way to completely clear the data caches on Consoles (I'm on Xbox, so I'm assuming PS4 does it a similar way for clearing caches). I searched the net with the phrase "Destiny 2 Nightingale error PS4 how to fix?" and got quite a few hits on issues with invalid DNS entry or ISP issues that prevent you from accessing Bungie's log in server, like this one: https://www.reddit.com/r/DestinyTechSupport/comments/9bbwf6/error_code_nightingale_unable_to_login_to_ps4/ Hope some of that helps...
Idk yeet your router out a window?