Same issue here. I wonder if it's a PS4 issue? I haven't tried typing in other games, but I know the PS4's bluetooth is kind of finicky and strange.

I've never had control input lag from the controller; even on remote play. But there's consistently a solid 1-2 second delay when typing with my Apple Bluetooth Keyboard. I sit 17 feet away from my PS4 (hooray for a 70" TV though!), so I haven't tried a wired keyboard. I guess, being new to the game; it hasn't bothered me too much. Like OP I type in excess of 120 wpm, and so I have to intentionally slow-up my typing when in FFXIV. I use the controller to play and the keyboard to chat or type commands.

It sure is frustrating; but even more frustrating to discover this is a year-old issue that seems to be plaguing a number of players. I play on PC as well and don't have the issue. (And I play with a controller/keyboard combo on PC for consistencies sake!)