Hanging Light Effect

It sounds like you almost had it, so I'll try to pick up from where I think you got to. The following uses a Light object and describes how to align it to your object for the animation.

Starting with your shape that you drew — Place it in a Group by itself (it can be top level or simply select the layer and type Command-Shift-G to create a new group with it inside.

If you haven't already done so, Add a Light object (use a Point light).

Drag the Light into the same group as the Bezier shape and line it up with the "bulb" or at the point at which you want to have the light emitting.

Select the Group that contains the Light and your Bezier shape and go to the Tools menu (underneath the Layers List pane [usually]) and select the Anchor Point tool. Position the Anchor Point at the fulcrum of the swing. Add the Oscillate behavior to the Group (I assume you know how to do that - Z-Rotation parameter) and set the parameters for the swing (Sine, about 24° Amplitude, Speed about 30).

The light will swing lined up to the Bezier shape and emit onto whatever background you provided.

All this is best done in 3D space, of course (Motion will convert when you add the Light). If you don't want to use 3D layers, skip the light, use a blur on the "light source" shape, and you can choose whether to add the Oscillate to your Bezier shape or the group.

HTH

.... addendum:

when you add a slight amount of Wobble to the x-rotation,

it adds a bit of realism

https://youtu.be/0RML1SYQQbc