^ This might be the root of the problem. I know wireless controllers sometimes go to sleep and FFXIV won't recognize them after re-activation. So if your keyboard goes to sleep, FFXIV might lose contact to it.

Though it seems a bit weird. If this setting is enabled per default, then other users (or at least those with the same keyboard) should have had the same issue. I stopped using Logitech keyboards because of weird glitches they had, like slowing down boot times or locking up, those were wireless though.