Perhaps I've got an audio wiring issue from the Dynon system when it's operating as the "master".
There are several levels at play here. Alert audio from the displays is normally tied together physically, and yours should be, according to you. Display alert audio and EMS audio travel different paths initially, so the loss of only alert audio at least narrows things down a bit. If the alert audio and EMS audio are not both routed through the G340, that might also help narrow things down. More on that later.
The first display powered on should be the only one outputting alert audio. That's what the master configuration does. Normally, this would mean the master configuration can't affect anything if both display outputs are tied together. However, as N941WR noted, the volume might be turned down on one display. It's not likely, but it's possible.
By the way, this isn't a twin engine plane with two EMS's, is it? That also disables the master display settings.
You can check continuity manually on each display. Go to SETUP MENU > SYSTEM SETUP > AUDIO SETUP, and select VOLUME CONTROL / TEST on each display in turn. This test bypasses the master setting. Or, as maartenversteeg mentioned, you can just power on one display at a time, which also bypasses the master setting. I would do that anyway, just to be sure. If your audio settings are correct, and both displays test correctly, you probably don't have a wiring issue, unless it's intermittent. That also confirms the alert audio outputs are tied together.
If the test fails on either display, you probably have a wiring problem. Since you have no problems with EMS audio, the problem must be occurring prior to the point where those two audio signals get tied together (electronically, not physically). In your case, I assume that's the GMA340, so check the settings there too. But before getting into the physical wire checks, swap the displays and run the audio tests again. If the problem switches with one display, then that display is likely the culprit. Then it would be time to contact Dynon. If the problem doesn't swap with the display, it's probably time to break out the multimeter. And don't forget to check pins along with the wires.
The tough part is that it's intermittent. All these tests need to be done while the problem is actually occurring, and that may not be easy to accomplish.