Camera & character math/logic

My character’s skeleton has a head bone, and an invisible bone sticking out in front of his face (which, for brevity, you could think of as a “socket” for the camera). The camera component itself is attached to a spring arm, which is targeting this camera bone. What I’d like to do is, while in 3rd person mode, have the spring arm’s target location rotate around the character’s head, based upon controller yaw input.

The goal here is to try and keep the character within view, but keep him out of the crosshairs in the center of the screen. So I figured I could simply offset the spring arm’s TargetOffset X value by like 100 or so, which only does the job if the camera isn’t pointing north (0 degrees) or south (180 degrees). So I figure if I could rotate the target around the character so that it’s always on one side, the character can never get in the way but will still always be visible on screen.

Am I overthinking this? What’s the best way to go about it?