Rogue Galaxy session 3

Knowing I needed to have the Factory Key to continue, I began searching.  I found two chests but neither contained the Factory Key.  There's also these locked chests that require keys I don't yet have to open, I can only assume I'll get these keys later on and then get to experience the fun of going back through all previous areas looking for chests to open with them.

So basically, grinding session.  I found that certain enemies dropped an item that a bunch of my characters needed for various things in their respective Revelation Flows, so I stopped to farm for a while.  Strangely, during this, the game told me after a fight that my inventory was full and I'd have to drop something.  Even though I have two free rows on the third tab and three entirely empty tabs of inventory space.  It seems like I can have a max of 104 inventory slots used currently.  Maybe that'll increase sometime soon?  Or maybe I should sell stuff or do item synthesis.  I dunno.

After going all the way back to the entrance to the place I'm in and working my way back to the door, I found zero unopened chests that didn't require keys.  I am now thoroughly confused as to how I'm supposed to get the Factory Key so I can continue with the game.  I'd look it up, but I'm really trying to do this first playthrough without any guides...