No, that's not how double-elimination works. If we were to have six teams into the double-elimination process, then a team will have to get two losses for it to be eliminated. The first loss will put a team into the losers bracket where it will have to fight against another loser team to stay in the competition.
The problem with this process is the number of teams. Since we have six teams, we'll have three winners and three losers in the first round. The match-up will be confusing after that, since the judges wouldn't know who should fight which.
I do believe eliminating three more teams to leave three for the Round Robin competition would be better, since they will have the chance to fight all the other teams.
EDIT: As for the poll, I don't see the problems of merging two members from two incomplete teams. They've made it that far. Surely it would be different if someone from a team that already lost would try to replace someone from the remaining teams.