Eternity and Steam Overlay

I like to use Eternity as my main source port when launching the Doom games on Steam.
Recently people have been joining my broadcast and complaining about not being able to see the Stream but hear the audio.
I know this isn't a source port designed for Steam and I am aware this is a very specific issue.
I was wondering though if someone here could think of a reason why this could be happening.
It has been bothering me for a while.

Are you running with the SDL GL 2D-in-3D video backend? Steam only works with/properly captures hardware accelerated APIs; emulated ddraw or GDI drawing are invisible to it.

Yes, I am.
Is there a way to launch without the system console window?