Let me see if I get this right. I've read though the thread a few times.
The pivot is rotating very slightly when opening and closing the knife? Is that the only play the pivot has? This is normal for my Emersons when I tighten or loosen the pivot screw, but that shouldn't happen when opening or closing the blade. Usually the friction and fitment of the pivot bolt in the handle scale and liners is enough to prevent it from rotating during opening (different story if using a screw driving to manipulate the pivot screw of course).
So to me, it seems like there's too much friction or play on the blade when it rotates around the pivot bolt. To test, take the blade and pivot out and then test to see how freely the blade can spin around it. I'll often use dish soap and a toothbrush to clean the pivot bolt and also the pivot hole in the blade. On rare occasions, very light sanding on the pivot bolt to remove any dings, damage, or rough spots. I try not to strip the coating on the pivot to bare metal because that can introduce some blade play if done inconsistently.
So try cleaning the pivot bolt and blade pivot hole. Then inspect the pivot bolt for damage. Reassemble the knife and play with different pivot screw tightness to see when the problem occurs or goes away.
When I tune my Emerson pivots, I usually put a finger on the bolt end to stop it from rotating for really fine adjustments. It's just how they are.