Unfortunately this is not possible "in a native way" on BGA. But of course as you suggest, you can do it by yourself with a series of tournaments.
One great possibility is to make use of the Swiss System tournament style. To give you some inspiration: You could set up a Harmonies two-stage tournament with Swiss System in the 1st stage, let's say 64 players distributed into 4 groups of 16 players. Each group would be its own Swiss System tournament. You could define that only group winners go to 2nd stage so that the 2nd stage would only include 4 players and could be made very short (e.g., by chosing single-elimination). The 2nd stage could be "just-for-fun", no need to abort the tournament. At the same time you can define that the first 5 players in each group of the 1st stage advance to another private tournament of another game, say, Azul. Of course you would need to invite those 20 players on your own.
Or you just go for a single large Swiss System tournament of 64 players without any 2nd stage, and you define that the top20 from it adnvance to another 2nd tournament.
In this 2nd tournament, you can again define some arbitrary progression rule, e.g., the best 4 in 2 groups of 10 make it to another tournament of Ticket to Ride.
Note that in the Swiss System, every player has only up to one running game at all times, each "matchday" is played after each other. If you chose round-robin style instead, all games start at the same time! The additional advantages of the Swiss system are that over time, players of similar skill are matched and you can create larger player groups. However, round-robin tournament have a tighter defined time frame and have a more controlled timing in that sense. Swiss System tournaments can drag on.
So, if to use Swiss style or round-robin style depends a lot on if you prefer one game after another of if you prefer simultaneous games running at once.
