The HDMI switches are likely not working because the Beam requires an HDMI-ARC connection, not standard HDMI signal.  Some switches can passthrough HDMI-ARC, but I’m not really sure that could help in your situation. The TV would be connected on the output side, while the Beam would be one of the inputs, as ARC (Audio Return Channel) data flows in the opposite direction of normal data flow.  You wouldn’t be able to get audio from your BT to the Beam that way.

I probably would look at using an optical switch instead.  Does your TV have an optical output?  If so you could use an optical switch to switch between TV/BT, then use the Sonos optical adapater between this switch and the Beam.  Of course, that means you won’t be getting any Dolby Atmos content though.

There are probably some better ways to do this. perhaps with an optical to HDMI adapter, HDMI switch, and HD Fury Arcana, but that gets rather complicated and expensive.

Perhaps the best solution is just to buy a Sonos Roam.  The Roam can be put in Bluetooth mode to receive bluetooth audio, and then be grouped with your Beam (only connected to TV) 

The HDMI switches are likely not working because the Beam requires an HDMI-ARC connection, not standard HDMI signal.  Some switches can passthrough HDMI-ARC, but I’m not really sure that could help in your situation. The TV would be connected on the output side, while the Beam would be one of the inputs, as ARC (Audio Return Channel) data flows in the opposite direction of normal data flow.  You wouldn’t be able to get audio from your BT to the Beam that way.

Why do you see the BT audio working differently?  The Sonos adapter converts TOSLINK to HDMI ARC, which I imagine would work (or not work, as the case has been) through a 1-in/2-out switch just as well as the TV audio.

Perhaps the best solution is just to buy a Sonos Roam.  The Roam can be put in Bluetooth mode to receive bluetooth audio, and then be grouped with your Beam (only connected to TV) 

ooh, clever!  It has the clear advantage of using Sonos as designed, no extra cables or switches.  As a BT receiver the Roam is by far the most expensive solution, but also the most elegant.  Thank you!

I would still like to understand what breaks when I add a HDMI switch between the Beam and these sources, if anyone has thoughts on that...

As melvimbe indicated, it would be ARC. Which is generated by the TV set after receiving a standard HDMI signal from whatever device is sending a signal . Most (not all) switches don’t really pass HDMI ARC, so using a standard switch with both the TV and the Bluetooth receiver connected via optical to the adapter, would likely be problematic. Especially as I think that Sonos adapter is powered by the Sonos device it’s normally connected to, and if you’re connecting it to a switch, it’s unlikely to be receiving power from the switch to do its work. 

Cost would be low to test this, though. You’ve already got most of the necessary equipment, just likely need to get an ARC passing switch, paying particular attention as to it being able to pass an ARC signal, as well as powering any devices connected to it, and then the non-standard wiring (TV’s ARC and Sonos adapter to inputs, Beam to the output of this switch).

Not something I’d do myself. I’d be much more likely to just do as melvimbe suggested, and use a Roam as a bluetooth bridge, with the recognition that I wouldn’t expect any lipsync if there’s a video associated with that audio. In fact, that would make things even more complicated, if there was any video associated with the bluetooth audio. 

Especially as I think that Sonos adapter is powered by the Sonos device it’s normally connected to, and if you’re connecting it to a switch, it’s unlikely to be receiving power from the switch to do its work.

good point

TV’s ARC and Sonos adapter to inputs, Beam to the output of this switch

this has been part of my confusion; the Beam’s port is an ARC input, designed to connect to a TV’s video input (and ARC output).  The ports on switches are described with respect to video I/O, so I’d assumed I wanted a 1-in/2-out switch, where the Beam was connected to the switch input port and the ARC audio sources were connected to switch output ports.  I don’t know how relevant this directionality is with respect to any active role the switch is taking, but since many switches don’t support ARC, and even those that do often limit it to particular input & output ports, there’s apparently a lot more going on than simple electrical switching.

I’d be much more likely to just do as melvimbe suggested, and use a Roam as a bluetooth bridge

agreed!  already ordered :-)

I wouldn’t expect any lipsync if there’s a video associated with that audio. In fact, that would make things even more complicated, if there was any video associated with the bluetooth audio. 

another good point, thank you!  Can you expand on what you mean by “more complicated”?

Inasmuch as you’d have to do a lot of figuring out how to get a device that can send video in one way, and audio another. Most phones/tablets/PCs can’t do that, some TVs have the ability. And then you’d need to deal with where any lag comes from. Is it from the originating device sending the signal, is it inherent to Bluetooth encoding? Sonos is pretty good about playing sound as quickly as it gets it, but not all devices send a signal as fast as they should. There’s just a lot of different combined areas for failure there. You’re cobbling together a frankenstein system, and then expecting it to run a marathon. Can it be done? Probably so. Is it worth the effort? Not to me, certainly, but that shouldn’t stop someone from trying it, with realistic expectations that it may work, or it may not.  I could probably pull a house with my little convertible Mazda, but is it wise, and will I have trouble, you bet. :)

I am not aware of any 1 in/2 out splitters that have ARC (Audio Return Channel) functionality, as it doesn’t really make sense for the purpose these devices are designed for.  They are not designed to send video from a single source to 2 display devices (sink).  While some of them have switching functionality, so they don’t send video to one of the displays, they aren’t really switches. Certainly not switches in the sense that you can select which sink device sends audio via ARC back to the source device.

My Roam was delivered yesterday, and it’s working great for playing Bluetooth audio and bridging that audio to other Sonos speakers.  Surprisingly, the latency is low enough that there are no obvious lipsync problems when watching videos from YouTube or Zoom.   Thanks again, @melvimbe & @Airgetlam!