So wait, the people who advance further than the first round don't get to play anymore after they're eliminated, but the first round eliminatees do?
There might be a losers bracket 2 but normally round 1 is where the biggest upsets happen and where people are still the most amped to play
Would make it impossible to have a "bracket", but you could have a window each week to allow losers of the current main bracket to join the losers bracket, and then shuffle them in with the winners from the previous round of the loser's bracket. That is:
Now 'til ~Tuesday: Signups for Round 1 of Losers' Bracket (Round 1 main tournament losers)
Week 1 (~Tuesday-Monday or whatever): : Round 1 of Losers' Bracket + Signups for Round 2 of Losers' Bracket (Round 2 main tournament losers)
Week 2: Round 2 of Losers' Bracket (winners of Round 1 LB + new signups) + Signups for Round 3 of LB (Round 3 main tournament losers)
etc.
Each week, everyone still participating has the same "record" between the two tournaments. Re-seed, or just do a big random draw every round.
tl;dr: Make it more like a Swiss format, and once you have two losses between the tournaments, you're out.