This might not answer your question but it might help because it may not address the HDMI output.
Canon has the EOS Webcam Utility Beta.
I downloaded it form my 7D II and have been using it as a webcam. If you are already using a webcam you might be able to substitute one of your bodies for the webcam.
It is listed for the 6D II on downloads at Canon's website.
The problem you will have is battery power for an hour and a half.

https://www.usa.canon.com/internet/p...vers_downloads