Sign up for tournaments in-game. This makes it easier for players to signup and easier for the moderators to manage list of players. It reduces human error when mods have to go through google doc signup sheets (which i assume is very time consuming as well) and gives the players direct confirmation of signing up.
Specialized tournament rooms. The custom lobby is usually a mess during tournaments in the first few rounds. If there are for example premade tournament rooms in alphabetic order at the bottom of the list it would be a lot easier for players to find their room. Inside this room all items should be available and fixed. There are many features within the rooms that could help with the moderating work, but I think putting them in order making it easier for players to find their room is rather important.