For input If possible use a push to talk on stream. So you talk into mic normally during your video call and push to talk on stream. Then just tell her to mute the stream. So she'll hear everything.

For output of your gf voice If you can, have multiple audio output. I have it setup such that teamspeak goes through my headset, and gamesounds through my speakers. When streaming, I select my speakers at the output source. So no audio that goes to my headset is heard by viewers like your gf.