I guess it's safe to assume that the game "updates" the universal inventory by removing the Cave items equipped as soon as we begin the Labyrinth, and only "reupdates" it with the ones obtained after reaching a Goal or exitting with Geomancer. Oh well! They really didn't make it any easy this time around.
And good to know it's possible to find some random Cave equipments besides the exclusive sets as cards along the way! This should help a little on the farming. Now, to find some free time.... u_u"