Hi there. I decided to make an account so I can chime in on this problem.
I have this exact same issue whenever I'm downloading a patch for the game. I rarely have any issues when I'm actually playing FFXIV, but for some reason, keeping a connection with PSN during a patch download is difficult.
My only advice is to keep trying. At certain points during the download process, the game will install a portion of what you've downloaded, so that if you are signed out, you'll resume where it last installed. I had to keep doing that until I got it broke down into smaller and smaller segments. It's definitely given me some anxiety as to how I'll manage to get future updates downloaded in a timely fashion. Hopefully it's something on their end that they can work out.