I think there's a problem with control points' handles but I'm not really sure how to fix it. I'm using Blender 3.4.1.
*** EDIT *** I'm attaching my .blend file if anyone would like to test directly on the problematic curve.
Also, some clarification: I would like to retain the catenary shape on all splines and the access to the handles of the control points after the duplication. The curve is a part of a bigger project (the .blend file is a simplified version), so I would also prefer to avoid converting to mesh and back if possible.
I tried to recreate the problem by hand and (if I understand correctly) the problem happens because as the curve gets offset and the distance between two control points gets smaller (or bigger), the vector handles stay the same, while actually they should be scaled down (or up) and moved closer (or further) to the original point on the XY plane.
Having said that, I have no idea how to fix this. I appreciate your help.