I got a bit confused during this since I was 100% certain I understood everything, but while my pieces were following the board, the distance they covered was not even. Turns out… I forgot to turn off proportional editing.
Instead of doing math for moving the pieces (who likes math?), I used our very nifty snapping tool and set it to “Edge centre”. Selecting a specific axis while hovering over a different facing axis centred the pieces perfectly in the middle of each face.
I don’t know if this is supposed to happen, but I noticed that while the linked pawns all show me their vertices at the same time in edit mode, the bishop doesn’t - But changes I make to one bishop affect the other regardless.