How do I prevent camera shaking when focused on gameobject?

I'm working on a game that uses a phone's gyroscope to look around and I need to use a sort of image stabilization when the user is looking at a specific game object. I've looked up various things that I thought would help, testing things like Quaternion.RotateTowards() and Quaternion.Lerp(), but they don't seem to serve the purpose of smoothing. I've also looked into maybe applying some sort of damping to the camera, but I'm not sure if that will help. Another note is that this needs to be a script for the camera alone and not effect the gyroscope viewing script. Any guidance would be greatly appreciated, thanks in advance.

The script is in C#