It's pretty easy to build auto reconnect capability in the client. The server will drop all its connections and go out of rotation, and the client will start a new connection and find the new one. If the switch happens fast enough then the user shouldn't even notice.

Along with the reconnect solution already mentioned, you can also decouple your Websocket and business logic layers using something like Pushpin: https://pushpin.org/. This allows you to deploy your business logic layer without disconnecting/reconnecting clients.

It is to be expected that LLM will make a decision on its own if it suspects any changes to the API. In any case, there is no time to fix the code during the game.