Further digging reveals this to be a hardware issue, not caused by the Just Flight PA-28.
I'm using a trim wheel from Flight Illusion, and unfortunately they've coded it to engage when AP is turned on... which works poorly for a plane with no AP Elevator Control... I've contacted their support to ask for an added option, to be used for planes like this. (they used to have that in their old software, but a "better" version has been released since)