Add this to your ux0:tai/config.txt for any application that requires PSN login to bypass it. Note this only bypasses login and the application might fail elsewhere if it requires PSN functions. It's useful for apps that require PSN but does not use it.
Note: After I made this I realized it still won't let you link your PS4 to a new Vita for remote play because that actually needs PSN. I haven't tested anything else.
The code is public domain and can be used in any way you want.