No sound through HDMI with Mac Mini Connected to Projection TV
I talked with the company that sold me most of my system (but not the Mac - just everything else) and we couldn't find a solution.  I called Metra, the people who make the HDMI switch and one of them suggested something I could get from Amazon.  It cost me $22.49 and fixed it.

I don't want to advocate for a particular brand, but what I found seems to not have a clear name yet.  Some are listed as HDMI feeders (the one I got that works is listed as an "HDMI feeader" [sic], which makes it easy to find.  Some are called HDMI EDID Emulators.  If you want to see examples, search for HDMI EDID changer (or emulator or anything close to that).

This is a small device that runs off the power from the HDMI cable.  Mine has a small post sticking out that looks like the post on a typical potentiometer.  There are 16 different positions and a chart on the back.  I picked "4k 5.1CH."  (The Sony projector is 4K, so the only thing that changed was the audio - which is, as I indicated, non-existent on the Sony projector.)

When I put the feeder in, between the Mac and the Metra HDMI switcher, the screen went blank for about 10-20 seconds, then the Mac display came back up.  Same when I changed the box to the setting I wanted.  It went blank for about 10 seconds, then the display returned.

I had a video playing on the Mac.  After the display came back on, with 2-3 seconds, the sound changed from being played on the Mac speakers to coming from my Sonos system.  (It may have changed with the display - I can't remember.)

So it's working okay.  The solution was the HDMI EDID emulator that edits the EDID from the projector to add the audio specs to it.