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)