So I think I found the solution for those like me trying to replicate their AE workflow:
The equivalent to mochaimport’s “stabilized precomp” is actually included out of the box in fusion and is called “perspective positionning” in the tracker operation tab.
It’s essentially the “reverse corner pin” I was looking for. What it does is take the four corners of the track and make them stick to the four corners of the shot.
The trick that got me is that it tracks the background but applies the transform to the foreground ONLY, and if you want to apply it on itself it’s not automatic, you have to pipe in the same footage to both ports (back and foreground).
You can then tweak the resulting “perspective corrected and stabilized” part of the image and reapply the movement and distortion by duplicating the tracker node and reapplying it in “corner positionning mode”
I hope it’ll save other people some time.