I am sorry, the GameNetwork currently doesn’t provide a mechanism to disable the log output manually. (The parameter you mentioned is used for another purpose). The only way to hide the messages is to do a “publish” build (stage in config.json), which disables all log output.
But you are right, the log for the game-network is quite massive – we will have a look if we can do something about that, e.g. adding a parameter like you suggested.