See images of it not showing up despite me not having it equipped: (See below images as to why it's locked.)
And before you ask, normally these locked items would show up but ASK you to confirm you want to use them. Instead, the Zig exchange doesn't even show them in the list at all. So I don't know why I have to unlock it first before it appears.
I just checked to make sure I was remembering correctly that this was how it normally is, and even on the JP version, with Zig's exchanges the weapons and units used for the exchange won't appear if they are locked. I can see where the confusion could stem from, though, since when grinding and affixing, locked items still appear on the list of fodder (though in that case you still get a message that they can't actually be used while locked), and the fact that you can use locked items in situations where the locked item wouldn't be lost/consumed, like Weapon Transmutation or Ring Skill addition.