I'd hazard a guess that it's somehow switching up the speaker config in the DirectX API or something. If you change the speaker config in the Windows API while still in the game, XIV may loose certain audio channels, or all sound until you reload the game. May be getting something similar to this since the game appears to only like 2CH or 5.1, and you may be swapping between them somehow when going from the analog to HDMI interface. Maybe if you could somehow synch the audio to the TV first with a media file or something?

I noticed this when I rebuilt my PC to Win8. The newer control panel for my Xonar DX has two drop boxes for speaker config. One synchs with what you set in the Windows Control Panel/Sounds snap-in, and the second one is what the hardware actually uses. Basically, it allows you to do virtual surround to different physical speaker combinations from 2ch/headphones up through 7.1 based on what you plug-in without Windows knowing you've changed speaker config and such. This is the only way I found to resolve an issue with voice-overs only going to my left surround speaker--if Windows was set to anything other than 2CH, voice-overs would only go to one speaker or not be heard at all. Had to keep reloading the game to test the different combinations until I found the mix that worked.