The projector does not have ARC. In fact, the only audio output from the device appears to be a stereo headphone jack.
I’d suggest pulling out the audio before the picture goes to the projector, using one of the Atmos enabled switches.
There is the HD Fury Arcana, the Feintech VAX04101k, and the OREI HDA-935. You can also order from those links from the companies directly, or check your local retailers for availability. Amazon may carry some, depending on location. At this point, there may be more manufacturers, I haven’t been keeping track.
Or, if you don’t care about Dolby Atmos, any standard HDMI switch with an optical port could then be connected to the Beam via the Sonos optical adapter, and you’d then get up to Dolby Digital.
Once you pull the audio from that device to the Beam, you could connect the video to your projector with the wireless HDMI, and not need the wireless HDMI device to support ARC/eARC.
Frankly, I’m still not enamored with this solution. It feels like you’re introducing fragility into a system that really isn’t designed for this use. In theory, it should work, in practice…who knows.