The ISP should know what's up with the PS4 but since I'm not them my workaround would be to look at a PC as an router. There's Virtual Router, Connectify and more to pull this off.
You say you have a router but that router should be handling the login so I'm not getting what the setup is here.
I brought my ps4 from the states to Aghanistan to use while I am here. I am using an ISP (ifone neda) that provides service to my area. It uses public access points with a Web page login with you're own user name and password. When I try to sign into psn I get an error code NW-31247-7 (A DNS server cannot be used, The DNS server did not respond within the time limit) however this was not always the case, when I first got here I had no problems but within the last 30 days it's been doing this. I was playing Destiny yesterday with absolutely no issues and it was working great. But i wanted to see if it was accutally fixed so i disconnected the Internet and tried to reconnect and I got the error code CE-33987-0 (cannot connect to psn within the time limit) so I tried logging out and logging back in and now I'm back to the first error code (I should have just stayed connected when it worked). The Internet on the ps4 works fine just trying to reach psn is my problem. I tried contacting playstation but they have yet to get back to me. And I cannot use their online chat because it is telling me I have to go through my region for support which doesn't offer chat, I can't switch regions (tried) because my PSN accout is a US account. If anyone has a fix please let me know. Here is some information that may be useful...
Using a cat5 cable connected to the router/ also tried wifi with same issue
Using a public access point
4mbs dl/2mbs ul
Nat type 2
Not using a proxy or vpn
Can easily log into psn on pc/phone/ps4 web browser
Stream netflix on ps4 fine (Internet works fine just no psn)
Tried reseting the router and unplugging it
Called my ISP they said it every thing was good on their end told me to check with psn
Thought it might be my service but other people are using PSN fine from here.
Reinstalled latest firmware 4.0
Tried different mtu settings
Can use ps4 remote play with my pc just fine
Tried multiple access points