Stop looking at a projector to handle your audio. I said this in your last post! Projectors aren't AV receivers, don't expect it to be one. It's insane. TVs aren't AV receivers either.

It's very weird you ended up with a stereo product without any video capabilities that you use instead of a proper AV receiver in your system. There are, literally, hundreds of options, at all sorts of price points, which do exactly what your DAC does, but were designed to handle both audio and video.

Still, what you need, to solve your problem, is a HDMI switcher with digital audio output. This type of product would be a HDMI switcher/audio extractor.
https://www.amazon.com/Version-SkycropHD-Extractor-Switcher-Converter/dp/B09N3G6BMK/ref=sr_1_6

I'm sure there are better or cheaper models out there, but if you want a stereo audio feed to your DAC over digital, then this is the type of product that can do it. You can hook up multiple devices to it, get a single stream of audio back to your DAC and call it a day.