Mmm thats weird, to unlock the double door you need to fight the boss Aiko (if i remember correctly) and pull a lever in the Blue door area, and then fight Feng in the red door area and also pull a lever.
I miss the part of pulling the lever in my firts game and then make me around the zone to much time xD. (Sorry for the bad english).
SPOILER FOR THE SLAVE PERK:
The foxgirl (Kotone?) also have a check list of progress (first-second will trial, aiko hard mode(don't remember 100% this) and final area boss hard mode, all of this tasks with the slave perk). This is not required to advance in the principal mission, buy give a juicy reward.