Camera Recommendation with specific requirements

cyberhiker

I am looking for an (in stock) camera or camcorder that has the ability to send audio+video over the HDMI port while recording. Preferably with an external mic input.  Basically I want the ability to record an event to SD card and send the captured signal (video+audio) via the HDMI port while recording.

I currently have a Canon T6i - while it sends the video over HDMI while recording the audio is not included in the HDMI signal and the video does not display on the built-in screen once the HDMI cable is connected (I guess the assumption is that the HDMI is primarily used for monitoring when in record mode).  In playback mode, both video+audio is sent but that is not what I need in this case.  I'm told most Canon cameras work this way so I'm guessing I need a different brand.

Additional info for background.   I'm currently recording a weekly event that is both live-streamed to a private Facebook group and recorded for later editing/upload to YouTube.  I now want to additionally send the 'live' video/audio to several other rooms with minimal lag.  I have an HDMI splitter that can do the distribution to multiple rooms.  (I looked at just having another device pull the live stream but there is too much lag by the time it goes to the cloud and comes back).  I'm trying to keep the setup as simple as possible - it will be operated by volunteers (otherwise I'd run separate vido+audio feeds into a laptop running OBS and distribute from there).