I haven't used mumble, (always encrypted communication), before but presumably the incoming audio is available on the headphone socket, (otherwise how could you hear it?).
I expect there is a setting to broadcast your audio locally too.
I spent a lot of effort on this about 5 years ago for a project I can't remember very now. Anyway something like the following should work:
http://www.startech.com/uk/AV/Converters/Video/usb-3-0-video-capture-device-hdmi-dvi-vga~USB3HDCAP
Seamless picture quality with USB 3.0 bandwidth enabling you to
capture 1080p video at 60fps
Record from virtually any device, with support for HDMI, DVI, VGA,
and Component video input sources
Supports 2-channel stereo audio (HDMI / RCA)

I've used lots of startech products, their support is good and they are reliable.