From my experience, You're on the right ideas.
In my experience, what I have done and played (Other DM used on me) is the smaller sessions with individuals. We had like 4-6 people in the story, so all people around the table waiting for a small encounter gets tedious real fast (and boring). So my suggestions would be (some You pointed out):
Do a simple 1v1 session, with the winning board (Have to battle everyone atleast once- "round-robin" tournament). Then every player can see, how every creature can fight another race. The one with most points, wins (As he proves he can defeat most opponents).
Regular bracket tournament (1 loss and You're out). This is a bit bad, because You get 1 unlucky situation and it's game over. I bet they want to test on all levels.
A very short timer. With 1v1, this should be simple. I did a timed battle (about 10 second limit) with 5 players and enemies. This put on a great rush. But if there are a lot of opponents, this gets very taxing on the players. So some mercy time should be added. But again, the format is tricky to think.
Put everyone in the arena and let them fight or do some strategy with timed moves. This is ofcourse bad, as there are 16 players. I bet some will lose motivation, so won't recommend this (Though would be fun to watch, could be end-game moment after the winner is known?).
There are a whole lot ways, but I personally would go with the 1st point.
I would improve possibly one way- You could invite some more DM-s to help manage the battles. Every player gets his own board and battles under the eyes of the judges (Can throw some story, that the overseers of the world or Gods want to see their champions fight). This way You could invite all 16 at the same time, let them battle and everyone can have fun at the same time (Some competitive atmosphere).
So basically 16 players, some judges, everybody fights everybody on different terrains (random) and winner is with most points.
One main thing is to give everyone equal chances- levels, gold, resources and other. Also, the terrain of the arena is very important (Some creatures have bonuses on some terrains, time, and such).
All the fights can be done with goal of 1 win, best of 3 (Recommended), or any other amount You'd like.
This is actually really cool idea, and I might use this in my group at one point also! :D