You could set it up before the tournament selection. Just use the seeding, without worrying about the teams in those seeds. In general, you need 18 games that are up for grabs, since 2^18 = 262,144 > 200,000. But you can pick those games based on seed, well ahead of the tournament.
As an example, once you reach the Final Four, there are 8 possible outcomes, since there are three games remaining.
200,000 / 8 = 25,000, and 25,000^(1/4) = 12.574.
To get at least 200,000 total, you’d pick the 13 most likely brackets for each of the four divisions. For the first round, all the higher-ranked teams win. For the next round, make the 4/5 match-ups equally likely, but follow the seeding otherwise. That get’s you four factors of 2
Once you’re in the Sweet 16, you also have eight possible outcomes for each division. So assume those are equally likely to get four factors of 8.
So total, you’ve got 2^4 * 8^4 * 8 = 524,288. This is more than twice what you need, so you could decide on one 4/5 seed where #4 always wins, to bring that down to 262,144.
Alternatively, you could have all the #1’s make it to the Final four, and work out the other outcomes.