My idea doesn't quite follow a strict double elim (this is impossible with RRs since someone is going to lose twice from winners, presumably without getting knocked out) but it only involves 1 round robin and follows a fairly normal bracket scheme.
1. Play the current Unwinners R7. This leaves 3 winners and 3 unwinners.
2. Play Round Robins for Winners R8 until someone goes 0-2. That person drops immediately to unwinners.
3. Play Unwinners R8.1 with 4 players like normal. This leaves 2 winners and 2 unwinners.
4. Play Unwinners R9 with 2 players like normal. This leaves 2 winners and 1 unwinner.
5. Skip Winners R10 because these people already played, and drop the 1-1 RR player from step 2 to unwinners. This leaves 1 winner and 2 unwinners.
6. Play Unwinners R10 with 2 players like normal. This leaves 1 winner and 1 unwinner.
7. Play Grand Finals like normal with winner from RR and winner from R10.
8. Optionally play Grander Finals.