
I have a similar issue, I'm using BMC with iPhone 15 Pro Max, USB-C out to HDMI dongle with USB Hub, in the USB Hub a RODE VideoMIC II. I have enable audio supervision in order to get sound through the HDMI out and record both video an audio on OBS. The problem is that when I select VideoMIC II as the audio input, since the VideoMIC has an audio output jack, for some reason it selects automatically the VIDEOMIC as the audio output for supervision, so no sound on the HDMI out. If you replug in the HDMI out, then input switches to the iPhone mic and sound is HDMI out. But as soon as you select the VideoMIC as the input, it also switches it as the output.
The only solution I have found is to use a separate audio cable for the sound from the VideoMIC audio jack to the MacBook audio jack and use this as input source.
I think this is probably a bug in the BlackMagic Camera IOS App, but I don't know how to report it to the developer team...