This would not work. Certainly not for any sort of home theater (Dolby Digital) output. 
 

The CONNECT (all generations) is a stereo device, and cannot accept or pass a home theater signal. It also has no digital input, so any stereo input would need to be the analog RCA jacks, which would incur the normal ~75 ms delay for processing/buffering the signal to play in sync on all Sonos rooms.

In fact, looking at the specs for this projector here, it doesn’t even output any type of home theater signal, which suggests that in order to use a Sonos home theater type system, you’d need to do what many others have done in the many threads about this type of issue in the Home Theater area of this forum, and separate the audio signal out before it reaches the projector. And the device that would need to be wired to that audio signal would need to be the Beam, and not a CONNECT.

Unfortunately, at this moment, Sonos doesn’t make any devices, new or old, that could accept a signal at the projector, and throw that signal to a Beam/Arc at the front of the room. Any installs to support a projection system are challenging, as you’d see in those many threads. 

Not sure what you mean with this:

”+ Sonos Connect Gen 1 - running optical to HDMI output adapter out of Connect and HDMI cable into projector (no eARC avail)”

The optical connection on the Connect is indeed an “out”, but it seems you need an “in” to get sound into the Sonos system, as you want it to send the sound to a Beam.

So, this will not work. If you’d use the analogue input on the Connect you would encounter the 70ms buffer so you’d create lag - and you’d lose surround. Also when sound is relayed from the Connect “room” in Sonos to the Beam “room” you lose surround.

A Sonos soundbar should be connected directly (via HDMI or optical) to an HDMI-capable TV or projector. So you need a direct connection to the Beam, either form HDMI-ARC if the projector can do, it or optical if it can do that. The Connect has no place in your set up.

The Play:1’s are rear speakers, not side speakers - there’s no side speakers in Sonos. They should be bonded to the Beam (and could not be bonded to the Connect because that does not do surround).

All your speakers are S1 capable, by the way if the Beam is a Gen 1.

Got it. That clarifies the role (no role) of the Connect and the conditions for a lag makes sense now. What would we do without y'all?? Thanks 106rallye & Airgetlam!

I will plan to get a Beam and expect some cross-room cable management. Either a long optical or a long HDMI (15-20') and use the Beam's included adapter on one end. (Sure hope one day Sonos makes a simple/low priced device to throw a signal across the room in the future.)

From your info "projector doesn’t even output any type of home theater signal... separate audio signal before reaching projector", here's splitter I hope to buy, allows 3x HDMI input, 1x HDMI output and optical SPDIF output:

https://www.amazon.com/iArkPower-Optical-Extractor-Splitter-Supports/dp/B01K7BZ1XC/

PS - With the proper keywords you provided, I found a post that summarizes nicely, including here for other readers:

https://en.community.sonos.com/home-theater-228993/projector-connections-sonos-and-av-sources-6837641