It looks to me more that the left hand IK is not working correctly / is delayed. Can you check whether the weapon moves with the right hand? It should be a child of the right hand.
If it's a child of the hand, it should move with it. This is how Unity works. And in your scene the weapon moves with the camera/body. The only thing I see is that the left hand is delayed in adjusting position.
The best way is always to compare your setup to the demo scene.