Depends, are you acting on the shotgun via animation or are you acting on the animation with the code?
binding the left hand IK to the shotgun pump would - regardless - move the hand to where the socket you take the position from is… the question becomes “is the hand following the shotgun or is the shotgun following the hand ?” and that’s basically what you need to work out to implement it. if you have it working on the shotgun with code you already know how much the hand slides and where…