Check your settings in OBS if your audio in source can be the firewire connection. Here is a link to how to do so -
If that doesn´t work my suggestion would be to have 2 audio sources.
In our church, we have a secondary mixer so we can have a separated mix for streaming, but you could have an audio workflow that could be:
- audio from the mixer to PC over firewire for recording, and from PC to mixer
- audio from an aux mix that can be transmitted via an audio interface or a sound card on your PC and your streaming software could use that as an audio source for the streaming