My first thought on what could be affecting unwanted offsets that vary per frame would be to check the priorities of your xpresso tags so that nothing is operating out of the desired sequence.
But my suggested solution would even be a bit different, and more in keeping with the 3d animation fallback of 'just fake it': For the parts that must maintain contact with other parts on this contraption, partially disconnect and directly parent them, or use a parent tag. Then hide the resulting gap somewhere else that is covered by other elements. So for instance, split the geometry of the hook from the spring, perhaps only even the part sticking out above the hole. Parent this disconnected hook geometry to the hole. Then hide this gap from the camera somehow. If the hidden gap is expanding and contracting a bit, ok, you won't see it. And be content with hiding these imperfections. They become nearly inevitable on any complex, interconnected machine like this.