It seems that this started happening when you updated your graphics card drivers, have you tried rolling back to the previous version of your graphics driver?
Tried to go back to the old driver, didn't work - It still pops up the same DirectX Fatal Error

A little tinkering has revealed that the FFXIV can run on the integrated graphics-card instead of the Geforce GTX 970m, if I want to run it in DirectX11. It just seems downright mad to me.

How often does this happen? Does it happen when you log in or after you've been playing for a bit?
It will happen on login - if the graphics setting is set to DirectX 11 and in Windowed or Borderless Windowed - It will also happen during the game if you change the setting too Windowed or Borderless Windowed or indeed alt-tab, after starting the game. This is only for DirectX11 - DirectX9 has had one stand alone error, that wasn't related to the 1100000f error and it hasn't repeated itself.

Double-check that your laptop isn't overheating and that the vents aren't being blocked.
No overheating on the system - Overheating on the system can cause slowdown of FFXIV (If I don't have maximum fan speed on), but doesn't cause the crash.