The Roam should be able to connect via Bluetooth, yes, assuming the projector’s Bluetooth transmitter is the appropriate codec. Note, however, that if the Roam is able to connect, it will be connecting to a stereo stream only. And you can’t stereo pair the Roam and the Move, but you can group them in the Sonos software, so each device will be playing both right and left streams. And if the pair of Moves were set up as a stereo pair, then they would be playing right and left as you desire. 

Finally, Sonos hasn’t been allowed to support Chromecast, but since Chromecast is how you’re connecting to the projector, and not how the projector connects to the Sonos, it shouldn’t matter.

This is not something I would support as a long time solution. It’s fragile, and Bluetooth by its very nature is only stereo, compressed, and the delay for the projector to get the signal into the Sonos system would make lipsync a potentially insurmountable issue. You would be much better to choose to connect using a Sonos device intended for video connection. The low latency input on any of those devices would be a big help with lipsync.

Sonos unfortunately doesn’t make a great solution for projectors, due to the fact that the projector needs to be in the rear of the room, and the speaker needs to be in the front, near the image. 

Thank you for your answer. I'm still considering Sonos Beam (plus 2x One SL) and wireless HDMI. Will this configuration work?

I’ve never tried wireless HDMI. Does it, and your projector, support ARC, the method by which Sonos receives its audio signal?

T

The projector has 2x hdmi. Chromecast 4 is plugged into one, and I plan to plug in wireless HDMI into the other. Beam has hdmi ARC as far as I know. 

Maybe there is another option to connect Sonos?

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. 
 

Thank you very much for your help