Gary,
You may want to test this again. The chime only comes on mine when the Connector stops transmitting. I can start and stop sounds from my phone (music, etc.) and the phone audio supersedes the TV Connector audio after about a second and then the TV audio cuts back in about a second after the phone audio stops. But no chime.
The only time I get a chime is when I pause or turn off the TV and after 45 seconds the HA chimes (for disconnection) I think because the TV Connector stops transmitting. I also get a chime when I go out of range.
But as I said at the beginning, with the TV playing, I can play sounds from my phone and I never get any chime as the TV audio cuts in and out, superseded by the phone audio when present. In fact, I can start playing a song on my phone, it supersedes the TV audio, and then 10 seconds later I pause the TV program, and I never even get the 45 second chime.
For reference, I have the old-style TV connector (without buttons on the top). Perhaps this is why I get a chime 45 seconds after the audio stops instead of the 20 seconds you reported. But other than that, the notification that causes the chiming should be HA-originated, and should be irrespective of which TV Connector one has.