OK, but does the winner of the winner bracket get to lose once in the championship game?

Logically it follows that if he never lost a game he'd move down to the losers bracket

Usually double elimination makes sense if there are "groups" like group A, B, C, D

so the guy who goes 2-0 can go on to the semi-finals and get a better seed
and the loser of the winner bracket can play the winner of the loser bracket and get a worse seed