Items from the non-breakpoint list ALWAYS have a chance of dropping. This doesn't necessarily mean that they will drop, but they can.
Items from the breakpoint list only have a chance of dropping when a boss has been 'broken'.
In limited dungeons you will either get one or two drops from bosses. One drop is always from the non-breakpoint list, the other is from the breakpoint list. Fear cores in limited dungeons seem to be pretty rare though, I've never seen one drop there yet. They drop a lot more frequently in trial dungeons.
I hope that clears things up a bit.