HDMI is digital, and the output of your aircraft comms is analogue. Also the HDMI socket is just for playback to a tv or whatever.
EDIT: I keep editing this post as I read through the manual. I'm pretty certain that your "HDMI" socket is actually a proprietory Sony analogue output.
You will probably have to change cameras, or take an audio recorder and sync the sound later. Somewhere on this forum is an earlier version of this thread with lots of audio recording ideas. Back when I was a BBC producer/cameraman I made adaptors for both fixed wing and helicopter comms sockets which worked and still work very well.
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