Actually folks, this likely isn't a proper solution.  The audio is streamed from the internet to your phone, and from there to the Chromecast device.  For a proper Chromecast solution, the audio should stream directly from the internet to the Chromecast device.

Please, please, please... Chromecast support.

Just basic support (passing the URL off to the device and a "stop" button) would be enough.

OK.  I have the answer to my own question.

Tasker: Misc / Send Intent

   - Action: com.vblast.xiialive.intent.action.PLAYBACK_RESUME

     (say)

   - Cat: none

   - Leave everything else empty except

   - Target: Service

And similarly for the other Intents.

Works well.

Thanks! Great feature!