A previous time that someone did this they tried to have the whole tourney over a single weekend. It was a clusterfuck, games started late, tons of people couldn't be found come game time, I remember most teams were comprised entirely of ringers, teams advancing through forfeits...
I think the best way to run it would be to stretch it out over the next few months, don't try to do all the games in a weekend or 2. Set the bracket and give a 3 week deadline, each team must schedule with their opponent and complete the game (or series) before the deadline.
Or you could schedule 1 matchup every Saturday night, same time as the weekly pugs. It would be exciting, but might not consistently work with every team members' schedules.
If you are not going to also pick alternates for each team (which we don't really have enough players to do that), you need to have a method for subs, because real life happens. I suggest that you keep track of what round every player is picked in, and subs are allowed only if they are from a later round than the person they are subbing for. If a team is more subs than original players, auto-forfeit.
Also need to hammer out some rules (double hof? boost core llama drama? dodge core? how to choose map and side)
+2cents