I've found there comes a point where your webcam can be too good.

The GR III/IIIx does not have HDMI out, however it does support the DisplayPort protocol where you can solve video output in various ways with adapters, do note the max video resolution output is 1080p. The main problem is that eventually it will overheat and turn off (within minutes). Video was and is not a strong suit with Pentax/Ricoh products unfortunately.