How can I aim with a laser sight attached to a gun on animated character?

Hello. I am using the Mocap Online Pistol Animations. I have a pistol model parented to the right hand of the character. I also have a "laser sight"(line renderer) attached to the tip of the pistol.

The issue is, when the character is walking, running, turning etc. (blending through it's animations). The pistol doesn't stay very "steady". The laser sight sort of "wobbles" around as the character is moving.

I tried setting IK targets to steady the hands. But ran into other issues.

What is the best solution to this problem?

Any help is appreciated! Thanks.