Alternatively, they could leave the game and rejoin whenever it was their turn. Any characters they had access to will remain, and you can give them the rights to them back when it's their turn to play. The button to leave the game is in the game view (not inside the game), in faint gray under the name. A bit of a hassle, but it'd do what you're trying to do - clear the chat between sessions, and make no more than one player is listed as a player at any time, and it'll be as anonymous as you can get.