I have suffered from the 20019 Error that Square Enix have been so quiet about since the Ps4 Beta's. I have downloaded the client more times then I care to remember. I have tried rebuilding my database again and again. Here is what finally worked, and may simply show my own stupidity.

After the initial download and install, FF ARR requests that you shut down you game and restart the client to install a small patch. I have always returned to my PS4 Dashboard and closed the game before restarting. I decided to shut down my ps4 completely and restart after it prompted me to restart my game. Viola! Install without issue. Try it if you haven't is my advice.