Seems like in order to figure out what the time commitment is, we need some idea how long a game is.
I'm personally in favor of KamP's 2 out of 3 matches constituting a game. In fact, he beat me to the suggestion, but I was going to throw it out there myself.

If we do that, for the teams there are, the games are likely to last by themselves up to 2 hours each. I can't commit to 6 hours a day, which would be 3 games.
(I know the lobster guys have looked at the teams, but have the rest of you? They're all looking like good teams. Of course the team that may soon be formerly known as the Knights of Ni look like the best team...)
We could just do 1 match. 2020 suggested each game would be about 20-30 minutes. In that case, playing 7 games is doable, because you're only looking at 3.5 hours, which I can do if scheduled carefully. Practically that probably translates to 4 or 4.5 hours, though. You have to wait for other teams to finish before you can play them, allow room for people to make mistakes, eat meals, go to the bathroom, etc.
Your_mom suggested we do a week of games, then a week off, then "finals" on the 16th. I'm not sure what he meant by finals even though he probably told me. Is it like the last round of the round-robin tournament? The round that decides final placing? Makes sense to me. That's what I've been calling the championship.

(every other team would still have to play, they have their own ranks to determine in that round)
We could skip the regular season part and just put teams in brackets arbitrarily. Yet another option for setting up the brackets is to have 2 members of each team meet on my server with one of the 4-team fortress maps and have them play a match. Top gets first, 2nd gets second, and so forth. Then set the brackets up from two matches where equal ranking in one match lines up with the other match at equal ranking. Can't use bye games to balance it if we lack a power of two, though.
I like mom's suggestion. Games would start on April 1, making March 31 the last day for team signup. Finish them up by April 10, then run finals the following weekend, which is 2020's target week. Then everyone's just committing to 1 game on that weekend, and the teams would schedule it amongst themselves, still. Maybe offer a grace period if a team doesn't make it, don't want to forfeit the final round after all.