It would take too much work and a "second copy" if the game running at the same time as the first would be A LOT harder than you think.
I get your reasoning, but there are probably other solutions to this problem. For example, if streamer mode is on names and colours are randomised each round. Or the streamers could just ban all colours and names of the streamers they are playing with from the text chat (stop people from saying anything about who it is)