On the PC it worked as follows (only wanted to connect via Genki and not via normal Bluetooth),
I hope it still works with the Switch.
I plugged in the Genki adapter, which was automatically recognised by Windows 10.
Plugged the small Genki Mic plug into the headphone jack of the PC.
Connected the Genki adapter to the headset with the left, blue button (I didn't try the red, right one).
After pairing, I pressed and held both Genki buttons (blue + red) together for 2-3 seconds.
After that, something blinked and the mic was recognised.
I have the Genki with the blue and red button, it should work the same way with the "grey" version.
Hope it helps

There is a small plug-in microphone. If I understand correctly, the switch can only use audio as an output via USB C and not as an input. But I have never really tested this. The adapter itself is great and works perfectly.