For anyone interested, I may have found a way to easily use your iPhone (not sure if this function is on other platforms) for typing in text chat. Basically,
Open the main PS app, select "Connect to PS4" and then "Second Screen". Press the little speech bubble at the top which then advises you that whenever PS requests you to enter text you can use your phone.
I've just tried this and it works instantaneously with mail so I don't see why text chat would be any different.
Hopefully it gets carried over and will save people typing with their controller or buying a keyboard.