How about taking inspiration from the Champions League and going for a Swiss system?
Eight groups of six, with 12 top seeds, 12 second seeds, 12 third and 12 fourth, i.e. two from each pot per group.
Then you play three games about one team from each of the other pots. So, if you're a top seed, you'd play one of the seconds, one of the thirds and one of the fourths.
Once each team has played their three games, the top of the table go straight through, with second playing third, i.e. 2A v 3B, 3B v 2A, etc
From there, straight knockout. Eight games instead of seven for 16 of the teams, but hopefully the final matchday should still have plenty riding on it.