So a few things to know.
Most Broadcast video capture and output cards (ie, HD-SDI, Matrox, BlackMagic, AJA, etc) do not have a GPU (a graphics processing unit) in the sense of being OpenGL compatible. I know that sounds sort of odd, but those cards work differently than the video cards used for 3D computing / gaming etc.
Syphon relies on the latter (a GPU), and needs OpenGL. Since the AJA/BlackMagic etc cards *do not* have a GPU or OpenGL support, special software has to be made to take anything from OpenGL (like Syphon) to be output and synced.
That sounds horribly complicated, but BlackMagic has an example that uses OpenGL to output to their cards, but no one has made a Syphon “enabled” version. That sounds like what you would need to solve your problem
Unfortunately I have pretty full plate, so I can’t whip up anything soon – but you might see if anyone elsewhere has done it.
You might also find commercial piece of software that has both Syphon support and Black Magic support, and could sort of “roll your own” solution via 3rd party software.