In my haste to resolder the wires that broke off on one of the attempts to stuff everything through the f-hole, I had attached the out wire from the piezo pot to the third lug instead of the push/pull section, so it was simply doing what I had wired it to do - operate as a normal pot.

The output of the piezo is nice and hot, so it blends in with the humbuckers very well, and works great on it's own. I've got some acoustic guitar impulse responses, and when I apply those to the piezo-only signal, it sounds pretty close to a real acoustic.