I can not use an external camera connected via BlackMagic in WebRTC. I tried it in my source code, web versions of Google Hangout, appear.in (whereby.com), etc in Chrome, Firefox, Safari. Anything worked.
Any idea how I can make it work at least for one browser?
You can't.
Unfortunately, most Blackmagic hardware has proprietary APIs and cannot be used as a standard device. This is by design. They only want to support software that fully utilizes the capabilities of their hardware.
The only product I know of that they make that isn't proprietary in nature is the "Web Presenter" which is standard UVC, allowing you to use standard OS-provided drivers and capabilities.