So, out of curiosity, are you running FFXIV through Steam? I'm not SquareEnix support, but I am way, way too familiar with the Steam controller code. And there *are* issues with the fact that FFXIV has its own specialized controller support *and* Steam has its own controller support, and the two can get into a bit of warfare.

By default, Steam grabs the controllers itself (to allow button remapping and macros and whatnot) and then exports the resulting controls as if it were another controller. But FFXIV's controller code does some funky stuff and sometimes—and I have no idea what determines when 'sometimes' is—does not get along with Steam's virtual exported controller. Specifically, it will sometimes try to connect to the real controller, and then Steam steals the real controller away from under it. Which causes FFXIV to go "welp, that controller's gone" and not to see the pretend one Steam is now exporting.

If you are running Steam, go to Steam's Big Picture settings (yes, it's buried in here even if you're not launching from Big Picture) and uncheck PS4 Configuration Support (if using a PS4 controller) or Xbox Configuration Support (if using an Xbox controller). That will disable Steam from grabbing those controllers to allow remapping; if that's the root cause for you, it should let FFXIV just hold onto the real controller without interruption.