That’s pretty specific but the idea is always the same.
The gun has bones because normally they animate.
Add a “sight” bone to the gun, parent it direct to the root bone.
Place the sight bone where it needs to be. The rest of the gun will follow.
A proper IK setup will move everything in place to actually hold the gun, so you shouldn’t have any further concerns with moving the character’s hands. It’s just the gun that is being moved.