You’d need one of two things.

First, and easiest, is a device that can take a normal HDMI signal, and split off an optical signal from it. This is fairly  common, there are lots of HDMI switches available that will take a signal from the HDMI stream, and pull it off to optical, which you would then connect to the adapter that comes with the Beam. There are lots of threads in this particular area of the forum about that kind of thing.

Slightly more difficult, and more expensive, is a device such as the HD Fury Arcana, which takes the HDMI signal and pulls it off into an ARC/eARC signal. Since the Beam can’t really handle eARC, it would default to the ARC settings, and would be, IMHO, a waste of money.

Using a Beam, you’ll never be able to process anything more than Dolby Digital, which, as it happens, is the most that the first option can handle anyway. 

Thanks for the suggestion…

This is what I thought I’d need to do after the first attempt (the HDMI 1>2 splitter) didn’t work. I’ll try this next. 

Of course, I have NO idea where that Sonos Optical Adapter is that came with my Beam almost two years ago, so I’m ordering another one of those also.

Again, thanks for the quick response!

The splitter should be a switch, most likely, and make sure it has an optical output, in addition to the HDMI output.

Yep, that's what I have ordered. Once I get it I'll report back so others can benefit from my experience 

I’m not sure what ports you have on your Projector, but just wondered if this extractor would ‘perhaps’ work for you? (It’s not clear if the output ports can be swapped, or not).

https://www.amazon.co.uk/gp/product/B08TX473Z8

On reading further detail, it looks like it probably won’t work after-all, as it mentions sending a ‘black image’ only to the audio output - oh well, never mind.‍♂️

Okay, here’s the latest update.

TL;DR - everything works, but can’t get 4K on my 4K monitor due to HDCP

The equipment:

Amazon Firestick 4K TV - the Firestick

Sonos Beam - the Beam

Sonos Optical/HDMI adapter - the adapter

Sonos One (2) for surround sound

LG 43” 4K Monitor (43UD79-B) - The monitor

I purchased this device from Amazon avedio links 4K@60Hz HDMI Audio Extractor, HDMI to HDMI + Optical Toslink SPDIF + 3.5mm AUX Stereo Audio Out, HDMI Audio Converter Adapter Splitter Support HDCP1.4 Full HD 1080P 3D - the splitter

The I/Os:

1. I plugged the Firestick into the splitter HDMI input. 
2. I plugged an HDMI cable from the splitter HDMI output to the monitor HDMI input (4K 60Hz input)
3. I plugged the adapter optical end into the splitter optical output
4. I plugged an HDMI cable from the adapter HDMI output into the Beam HDMI input

I added the Optical input through the Sonos Beam setup, and away we went.

The details:

When I first attempted to play 4K content from Amazon Prime I got the following error message:

This TV HDMI port does not support HDCP 2.2. Content will play back at the next

best support resolution. Try connecting to other HDMI ports on your TV.

I knew the TV DID support HDCP 2.2, so I checked the splitter and lo-and-behold, it doesn’t. That’s my own fault for not checking that I needed 2.2...when the description of the splitter clearly says HDCP 1.4.

So I’m ordering a different model splitter that supports HDCP 2.2, but at least I know now I can make this work. 

I’ll report back again when the NEW splitter comes in. Hopefully that will be the last post. <grin>

Okay. I hope this is my final post about this, but it looks like it has been resolved as best as it can given the limitations of my LG monitor.

I purchased this splitter from Amazon:

NEWCARE 4K@60HZ HDMI 2.0b Audio Extractor, HDMI Audio Converter Adapter,HDMI to HDMI + Optical Toslink SPDIF + 3.5mm AUX Stereo Audio Out, Support HDMI 2.0b, HDCP 2.2, RGB 4:4:4,3D (Metal Black)

And hooked it up much as before.

• Firestick into splitter HDMI input
• Sonos optical adapter to HDMI cable, splitter optical output to Sonos Beam via HDMI cable
• HDMI cable from splitter HDMI output to input on monitor

It STILL appeared that I wasn’t getting 4K output to the monitor based on the Expanse series showing “HD” when starting the episode, which in turn led to a lot of troubleshooting with other monitors, Fire TV devices, cables, etc.

Finally, I googled how to check output from Fire devices.  DOH! There is a “Developer” menu option that you can enable on Fire devices that will show you the video resolution output of  the device.

I downloaded an app that made launching the developer menu easier than the “hold x + y button down for 3 years, then do a back-flip” type of thing.

This produces an overlay of technical info on whatever you’re watching on your monitor. It turns out that the FireTVs often start at one resolution and then adapt as they’re able to buffer. That’s why when watching “the Expanse” my monitor would show “HD” when starting, which made me believe I wasn’t getting 4K. However letting it run for 3-4 minutes I was able to watch the realtime resolution change from HD to 4K as indicated by the overlay. The “info” when pausing after several minutes would also change to “Ultra HD”.

SUCCESS!

For more in the Developer Tools menu, see here.

The app for easily launching the Developer Tools menu is “Developer Tools Menu”

Thanks for the advice and suggestions. I hope that this might help others who attempt this journey. If anyone has any questions, feel free to ask!