After the greater part of a day, I solved it! The issue did lie with my camera component. There is one setting that I enabled to get my old mouselook script to work, that, upon disabling, allowed me to rotate the camera on leaning. I can still use the leaning script that was posted in the question without any changes being made.
Here is my fix:
On the camera component, “Use Pawn Control Rotation” should be disabled.
This needed to be changed in my mouse look script (for only the Y axis)
My character can now lean around corners smoothly!