Re: Cannot play any part of the game
After you bought the PlayStation + membership for your PSN account, you are getting still the same update error message when you install the game, correct? You tried the most of the general PlayStation 4 troubleshooting steps, had you any other issues aka do you own any other disc based games for the console yet? It should not be an issue on an account level if you can't update the game without an error message, have you linked your console via wired rather than wireless connection to the internet and tried to power-cycle your router?
To do so, unplug your router from the power-source for about 30 seconds and connect it back to the power supply after. It might even be that the disc is damaged somehow but I guess you bought a new game without any scratches on the surface? May I ask where you bought the game and if there is any chance to borough a PvZ:GW gamedisc from a friend of yours?
Sorry that we are tripping in the dark here but with console troubleshooting, especially after you tried already many steps to solve your issue, we need to go the trial and error route.
Please keep us updated.
Basti