How to make the flash player on the site output video through an overlay?

Question

Windows

Hello.
It is necessary to achieve the output of the video broadcast on the web page from the flash player to the overlay. That is, only the broadcast picture itself should go to the second monitor (even with video controls). The trouble is that the flash player with the overlay does not work, so there are no regular ways to solve the problem.
The conditions are such that the broadcast must go directly through the player, so using the browser cache or getting the direct address of the broadcast with opening in vlc, for example, is unfortunately not suitable. Only hardcore.
I understand that it all looks rather wild and strange, but I will still be grateful for any useful information.