The best way I found to stabilize the bobbing is to manipulate the camera location based on previews frame and distance.
To avoid puking you want to disable all small movement occurring within a frame.
Let’s say a distance of 10 or greater is acceptable as it will signify the camera is moving along with the character.
The end result of setting up something similar is a responsive camera that moves with the character but doesn’t (assuming your animations aren’t really bad and move the head more then 10cm per frame) Bob or twitch along with the animation despite being hooked to the head bone.
For further development I create a Camera bone so that i can actually attach the camera and manipulate the bone directly. This proved to work much better in the end as the bone movent is a little more straightforward than socket placement or camera component local values.
The obvious benefit is that you can also create scripted views by manipulating the bone for things like cinematics without having to say tilt the character’s head in a way that’s not anatomically possible just to look up or down… and still allow for transitions between 3rd and 1st.
As a note on that, child actors are the way to go so that you may actually use the built in camera transition system.