I haven't set up the official Google Authenticator app in a while, but what I have done in the past is to seed the watch with the same value as the phone app. I

I haven't set up the official Google Authenticator app in a while, but what I have done in the past is to seed the watch with the same value as the phone app. It marginally increases the risk, as I now have two devices that could be stolen, but the convenience was worth it to me. That may require using a third party app on the phone as well as the watch, though. I'm not sure if the official Google Authenticator app still makes this practical.