Had to do this twice in the past six months. Once on my PC when I decided to reinstall the game after a reformat, and once on my laptop when I bought the game for my MacBook while at my aunt's house over Thanksgiving. I have Charter, she has Wow as an ISP.
I've had to switch between the ISP assigned DNS servers, Google public DNS, and any third option from a list of public DNS servers. Once you run into the issue on a particular patch file, the launcher will close automatically. Then you switch the DNS settings on your machine, and try again. Let it patch until it fails, and keep doing this until you get it. For the most part I was able to switch between Google DNS and ISP DNS servers, but there were a handful of patches that failed on both and I had to introduce a third set of DNS servers. In my experience patching the game fully twice, I've never had to use a fourth set of DNS servers, but again, I've only had to do it twice. It's annoying because you have to babysit but you should be able to get fully up to date.
I should note that when I was home I updated the router DNS settings, but at my aunt's house I just did it on my laptop. I'm guessing you can set the DNS settings on the PS4 directly although I've never done it myself, but if not the router level works as well.