Can I detect if the camera is looking to the left or right of the current heading?

[_Aka_]

I am assuming that your camera is on the spline.
What you want is the sign of the signed angle between the spline tangent and the camera forward.
try this

Code:

var projectedCameraForward  =  Vector3.ProjectOnPlane(cameraForward, splineOrientationUp); // You should handle the cases where projectedCameraForward is Vector3.zero;

FluffyUnderware.DevTools.Extensions.Vector3Ext.AngleSigned(projectedCameraForward, splineTangent, splineOrientationUp)