I found a very nice solution to the OP's question. There's a company named Rolls based in Lake City, Utah that makes a device called the "DU30B Audio Ducker" (available from many places, including Amazon). I'm using it to dual-source my TV audio and an Echo Dot into one input of my (admittedly legacy, but nice) stereo.
The TV goes into the LINE input, it's the "ducked" device. The (output from the) Echo goes into the AUX input, it's the "ducking" device. So normally, the TV just plays its audio as usual. But if I want to use the Echo, I just speak the wakeword, and it silences the TV and takes over the audio. This has a couple of nice side benefits. One is that silencing the TV makes it easier for Alexa to understand my speech. Another is that I can play audio from the Echo (e.g., streaming music) without having to turn on the TV. The Rolls DU30B switches seamlessly and silently between the inputs, with adjustable sensitivity and delays.
I use a Fire TV cube to control the TV itself, and the Echo Dot interacts with and controls it - again, seamlessly. I set the wakeword on the Fire TV cube to something different so I'm always speaking just to the Echo.
I read up on the MCM Custom Audio device referenced above, and apparently it uses a loud-ish mechanical switch that is annoying. I also tried the Echo Link and found that it didn't have the adjustments or functionality to work (officially, a TV input is not supported). The Rolls device plus an Echo Dot gives me everything I need, and is less expensive than the Echo Link.