Hi Peter,
In short, it's not possible to copy a custom bezier spline on multiple objects.
But I have to add some explanation to this.
So whenever you create a custom bezier spline, all the values are being hardcoded. So when you copy the spline onto a different object, it will copy the same values as well. So like you said, that's not a solution.
If you need an animation that you can reuse, you could look into Stagger Animations. These are animations that you can reuse on multiple objects. Originally these were designed for character animations, but you can also use them on groups as well. They don't work on quads for example, but if you put it into a group you can use it on a group.
However, a little warning, there is no way to use the KeyGraph editor on Stagger Animations. So you can reuse them but they're not as practical as custom animations.
Using the API you could also script the splines but as far as I know, only the TCB-splines are available in the API as SetKeyFrameValueTCB. I don't think the Custom Beziers can be programmed. But to be honest, I haven't scripted an animation in a long time.
Best regards,
Kenneth
#XPression