I'm not entirely sure that'd work very well, since the Chromecast mirroring function has some pretty high latency.

If you want to stream Dolphin to your TV, a cheap option is to use Steam's in-home streaming feature. The Steam Link is a dedicated piece of hardware for this which I think is about $50 and will work with existing controllers or the Steam controller. You can also just use any existing laptop (windows, mac, linux) with a sufficiently powerful cpu for stream decoding and use it as an in-home streaming client. I've messed around with a lot of solutions for streaming from my PC to living room TV, and Steam's solution blows just about everything else out of the water.