To my understanding, the stream port on the mixamp should be outputing a signal which is a mix of the game audio it expects to get from the optical input, voice comms which it expects on the usb and finally audio from the mic which of course comes from the headset. A cable from that port into the live gamer HD audio in should then get you what you want.
Thus, the audio device you should set to be used in OBS is the "audio in" of the capture card. If you're not using the optical in for game audio but just the usb interface, you might as well choose the mixamp audio device as your audio device in OBS and the headset mic as the "microphone/auxiliary" device.
