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You can tab cast Amazon Instant Video from Chrome. I've been successfully using it for months.

The downside, though, is that the video is still playing on the laptop simultaneously.

Yes, but you can dim down the LCD and the audio is piped to the chromecast. It's a 'hack', but it does work.

And devouring your battery as it does so. Native Amazon Video support is a must-have for me to use it full-time.

tab casting is (still) not supported in linux

I tab cast in Linux fine.. needed the latest build of Chrome at the time currently running 27.0.1453.12 dev.