the ATEM can not connect to Dante
Logic and ProTools can be used to process livestream mixes.
I was trying to say that all you need is in the GLD already
there is no real need for the Logic route
and there are tools available to do audio processing without the need of a real sequencer software
otherwise it is possible to feed the mix out the GLD into any interface (some of them have S/PDIF)
and back from Logic to the ATEM
latency should be no issue since the video part needs way more processing time than the audio
For the Dante Card you need an additional Dante Virtual Soundcard for the computer on which Logic is running.
the same applies to the Waves solution but the software is free of charge
Dante Virtual Soundcard costs $ 29.99